:: REALSET2 semantic presentation

definition
let IT be LoopStr ;
attr a1 is zeroed means :Def1: :: REALSET2:def 1
for a being Element of IT holds
( the add of IT . a,the Zero of IT = a & the add of IT . the Zero of IT,a = a );
attr a1 is complementable means :Def2: :: REALSET2:def 2
for a being Element of IT ex b being Element of IT st
( the add of IT . a,b = the Zero of IT & the add of IT . b,a = the Zero of IT );
end;

:: deftheorem Def1 defines zeroed REALSET2:def 1 :
for IT being LoopStr holds
( IT is zeroed iff for a being Element of IT holds
( the add of IT . a,the Zero of IT = a & the add of IT . the Zero of IT,a = a ) );

:: deftheorem Def2 defines complementable REALSET2:def 2 :
for IT being LoopStr holds
( IT is complementable iff for a being Element of IT ex b being Element of IT st
( the add of IT . a,b = the Zero of IT & the add of IT . b,a = the Zero of IT ) );

definition
let L be non empty LoopStr ;
redefine attr a1 is add-associative means :Def3: :: REALSET2:def 3
for a, b, c being Element of L holds the add of L . (the add of L . a,b),c = the add of L . a,(the add of L . b,c);
compatibility
( L is add-associative iff for a, b, c being Element of L holds the add of L . (the add of L . a,b),c = the add of L . a,(the add of L . b,c) )
proof end;
redefine attr a1 is Abelian means :Def4: :: REALSET2:def 4
for a, b being Element of L holds the add of L . a,b = the add of L . b,a;
compatibility
( L is Abelian iff for a, b being Element of L holds the add of L . a,b = the add of L . b,a )
proof end;
end;

:: deftheorem Def3 defines add-associative REALSET2:def 3 :
for L being non empty LoopStr holds
( L is add-associative iff for a, b, c being Element of L holds the add of L . (the add of L . a,b),c = the add of L . a,(the add of L . b,c) );

:: deftheorem Def4 defines Abelian REALSET2:def 4 :
for L being non empty LoopStr holds
( L is Abelian iff for a, b being Element of L holds the add of L . a,b = the add of L . b,a );

E29: now
consider x being set ;
set B = {x};
E17: x in {x} by TARSKI:def 1;
consider og being BinOp of {x};
E18: og . [x,x] = x
proof
E19: ( dom og = [:{x},{x}:] & rng og c= {x} ) by FUNCT_2:def 1, RELSET_1:12;
[x,x] in [:{x},{x}:] by , ZFMISC_1:def 2;
then og . [x,x] in rng og by , FUNCT_1:def 5;
hence og . [x,x] = x by , TARSKI:def 1;
end;
reconsider ng = x as Element of {x} by TARSKI:def 1;
E20: for a, b, c being Element of {x} holds og . (og . a,b),c = og . a,(og . b,c)
proof
let a be Element of {x}, b be Element of {x}, c be Element of {x};
( a = x & b = x & c = x ) by TARSKI:def 1;
hence og . (og . a,b),c = og . a,(og . b,c) by ;
end;
E24: for a being Element of {x} holds
( og . a,ng = a & og . ng,a = a )
proof
let a be Element of {x};
a = x by TARSKI:def 1;
hence ( og . a,ng = a & og . ng,a = a ) by ;
end;
E25: for a being Element of {x} ex b being Element of {x} st
( og . a,b = ng & og . b,a = ng )
proof
let a be Element of {x};
take ng ;
thus ( og . a,ng = ng & og . ng,a = ng ) by TARSKI:def 1;
end;
for a, b being Element of {x} holds og . a,b = og . b,a
proof
let a be Element of {x}, b be Element of {x};
( a = x & b = x ) by TARSKI:def 1;
hence og . a,b = og . b,a ;
end;
hence ex A being non empty set ex og being BinOp of A ex ng being Element of A st
( ( for a, b, c being Element of A holds og . (og . a,b),c = og . a,(og . b,c) ) & ( for a being Element of A holds
( og . a,ng = a & og . ng,a = a ) ) & ( for a being Element of A ex b being Element of A st
( og . a,b = ng & og . b,a = ng ) ) & ( for a, b being Element of A holds og . a,b = og . b,a ) ) by , , Def1;
end;

registration
cluster non empty strict Abelian add-associative zeroed complementable LoopStr ;
existence
ex b1 being non empty LoopStr st
( b1 is strict & b1 is Abelian & b1 is add-associative & b1 is zeroed & b1 is complementable )
proof end;
end;

definition
mode Group is non empty Abelian add-associative zeroed complementable LoopStr ;
end;

definition
let IT be 1-sorted ;
attr a1 is trivial means :: REALSET2:def 5
the carrier of IT is trivial;
end;

:: deftheorem Def5 defines trivial REALSET2:def 5 :
for IT being 1-sorted holds
( IT is trivial iff the carrier of IT is trivial );

Lemma35: for S being 1-sorted holds
( S is trivial iff for x, y being Element of S holds x = y )
proof end;

registration
cluster trivial 1-sorted ;
existence
ex b1 being 1-sorted st b1 is trivial
proof end;
end;

E41: now
let A be non trivial set ;
let od be BinOp of A, om be BinOp of A;
let nd be Element of A;
let nm be Element of A \ {nd};
let nm0 be Element of A;
assume E17: nm0 = nm ;
set F = doubleLoopStr(# A,od,om,nm0,nd #);
thus not doubleLoopStr(# A,od,om,nm0,nd #) is trivial
proof
now
take nd = nd;
take nm0 = nm0;
not nm in {nd} by XBOOLE_0:def 4;
hence nd <> nm0 by , TARSKI:def 1;
end;
hence not doubleLoopStr(# A,od,om,nm0,nd #) is trivial by ;
end;
thus doubleLoopStr(# A,od,om,nm0,nd #) is strict ;
end;

registration
cluster strict non trivial doubleLoopStr ;
existence
ex b1 being doubleLoopStr st
( not b1 is trivial & b1 is strict )
proof end;
end;

definition
let A be non trivial set ;
let od be BinOp of A, om be BinOp of A;
let nd be Element of A;
let nm be Element of A \ {nd};
func field c1,c2,c3,c4,c5 -> strict non trivial doubleLoopStr means :Def6: :: REALSET2:def 6
( A = the carrier of it & od = the add of it & om = the mult of it & nd = the Zero of it & nm = the unity of it );
existence
ex b1 being strict non trivial doubleLoopStr st
( A = the carrier of b1 & od = the add of b1 & om = the mult of b1 & nd = the Zero of b1 & nm = the unity of b1 )
proof end;
uniqueness
for b1, b2 being strict non trivial doubleLoopStr st A = the carrier of b1 & od = the add of b1 & om = the mult of b1 & nd = the Zero of b1 & nm = the unity of b1 & A = the carrier of b2 & od = the add of b2 & om = the mult of b2 & nd = the Zero of b2 & nm = the unity of b2 holds
b1 = b2
;
end;

:: deftheorem Def6 defines field REALSET2:def 6 :
for A being non trivial set
for od, om being BinOp of A
for nd being Element of A
for nm being Element of A \ {nd}
for b6 being strict non trivial doubleLoopStr holds
( b6 = field A,od,om,nd,nm iff ( A = the carrier of b6 & od = the add of b6 & om = the mult of b6 & nd = the Zero of b6 & nm = the unity of b6 ) );

definition
let IT be 1-sorted ;
redefine attr a1 is trivial means :: REALSET2:def 7
for x, y being Element of IT holds x = y;
compatibility
( IT is trivial iff for x, y being Element of IT holds x = y )
by ;
end;

:: deftheorem Def7 defines trivial REALSET2:def 7 :
for IT being 1-sorted holds
( IT is trivial iff for x, y being Element of IT holds x = y );

consider x being set , y being set such that
Lemma49: x <> y by VECTSP_1:78;

set Z = {x,y};

Lemma51: x in {x,y}
by TARSKI:def 2;

Lemma52: y in {x,y}
by TARSKI:def 2;

for s being Element of {x,y} holds {x,y} \ {s} is non empty set
proof end;

then reconsider A = {x,y} as non trivial set by REALSET1:6;

reconsider nd = x as Element of A by TARSKI:def 2;

Lemma54: for t being set holds
( t in [:A,A:] iff ( t = [x,x] or t = [x,y] or t = [y,x] or t = [y,y] ) )
proof end;

Lemma58: [:A,A:] = {[x,x],[x,y],[y,x],[y,y]}
by Def2, ENUMSET1:def 2;

set x1 = [[x,x],x];

set x2 = [[x,y],y];

set x3 = [[y,x],y];

set x4 = [[y,y],x];

set F = {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]};

Lemma63: for p being set st p in {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} holds
ex q, r being set st [q,r] = p
proof end;

for q, r1, r2 being set st [q,r1] in {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} & [q,r2] in {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} holds
r1 = r2
proof end;

then reconsider od = {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} as Function by , FUNCT_1:2;

for t being set holds
( t in [:A,A:] iff ex r being set st [t,r] in {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} )
proof end;

then Lemma69: [:A,A:] = dom od
by RELAT_1:def 4;

then Lemma70: [x,x] in dom od
by Def2;

Lemma71: [[x,x],x] in od
by ENUMSET1:def 2;

then Lemma72: od . x,x = x
by , FUNCT_1:def 4;

Lemma73: [x,y] in dom od
by Def2, Def3;

Lemma74: [[x,y],y] in od
by ENUMSET1:def 2;

then Lemma75: od . [x,y] = y
by , FUNCT_1:def 4;

Lemma76: [y,x] in dom od
by Def2, Def3;

Lemma77: [[y,x],y] in od
by ENUMSET1:def 2;

then Lemma78: od . [y,x] = y
by , FUNCT_1:def 4;

Lemma79: [y,y] in dom od
by Def2, Def3;

[[y,y],x] in od
by ENUMSET1:def 2;

then Lemma80: od . y,y = x
by , FUNCT_1:def 4;

then Lemma81: for t being set st t in [:A,A:] holds
od . t in A
by Def1, , Def2, , , ;

set s2 = [[x,y],x];

set s3 = [[y,x],x];

set s4 = [[y,y],y];

set D = {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]};

Lemma86: for p being set st p in {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} holds
ex q, r being set st [q,r] = p
proof end;

for q, r1, r2 being set st [q,r1] in {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} & [q,r2] in {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} holds
r1 = r2
proof end;

then reconsider om = {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} as Function by , FUNCT_1:2;

for t being set holds
( t in [:A,A:] iff ex r being set st [t,r] in {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} )
proof end;

then Lemma87: [:A,A:] = dom om
by RELAT_1:def 4;

then Lemma88: [x,x] in dom om
by Def2;

[[x,x],x] in om
by ENUMSET1:def 2;

then Lemma89: om . [x,x] = x
by , FUNCT_1:def 4;

Lemma90: [x,y] in dom om
by Def2, Lemma29;

[[x,y],x] in om
by ENUMSET1:def 2;

then Lemma91: om . [x,y] = x
by , FUNCT_1:def 4;

Lemma92: [y,x] in dom om
by Def2, Lemma29;

[[y,x],x] in om
by ENUMSET1:def 2;

then Lemma93: om . [y,x] = x
by , FUNCT_1:def 4;

Lemma94: [y,y] in dom om
by Def2, Lemma29;

[[y,y],y] in om
by ENUMSET1:def 2;

then Lemma95: om . [y,y] = y
by , FUNCT_1:def 4;

then Lemma96: for t being set st t in [:A,A:] holds
om . t in A
by Def1, , Def2, , , Lemma35;

then Lemma97: om is BinOp of A
by Lemma29, FUNCT_2:5;

Lemma98: A \ {x} = {y}
by , ZFMISC_1:23;

then Lemma99: [:(A \ {x}),(A \ {x}):] = {[y,y]}
by ZFMISC_1:35;

Lemma100: for t being set st t in [:(A \ {x}),(A \ {x}):] holds
om . t in A \ {x}
proof end;

reconsider nm = y as Element of A \ {nd} by , TARSKI:def 1;

reconsider od0 = od as BinOp of A by Def3, , FUNCT_2:5;

reconsider om0 = om as BinOp of A by Lemma29, , FUNCT_2:5;

Lemma103: for a, b, d being Element of A holds od0 . (od0 . a,b),d = od0 . a,(od0 . b,d)
proof end;

Lemma105: for a being Element of A holds
( od0 . a,nd = a & od0 . nd,a = a )
proof end;

Lemma106: for a being Element of A ex b being Element of A st
( od0 . a,b = nd & od0 . b,a = nd )
proof end;

for a, b being Element of A holds od0 . a,b = od0 . b,a
proof end;

then Lemma107: LoopStr(# A,od0,nd #) is Group
by , , , , , , ;

reconsider om1 = om as DnT of nd,A by , , REALSET1:def 8;

Lemma109: for B being non empty set
for P being BinOp of B
for e being Element of B st B = A \ {nd} & e = nm & P = om1 ! A,nd holds
LoopStr(# B,P,e #) is Group
proof end;

Lemma112: for a, b, d being Element of A holds
( om0 . [a,(od0 . [b,d])] = od0 . [(om0 . [a,b]),(om0 . [a,d])] & om0 . [(od0 . [a,b]),d] = od0 . [(om0 . [a,d]),(om0 . [b,d])] )
proof end;

definition
let IT be doubleLoopStr ;
attr a1 is Field-like means :Def8: :: REALSET2:def 8
ex A being non trivial set ex od being BinOp of A ex nd being Element of A ex om being DnT of nd,A ex nm being Element of A \ {nd} st
( x = field A,od,om,nd,nm & LoopStr(# A,od,nd #) is Group & ( for B being non empty set
for P being BinOp of B
for e being Element of B st B = A \ {nd} & e = nm & P = om ! A,nd holds
LoopStr(# B,P,e #) is Group ) & ( for x, y, z being Element of A holds
( om . [x,(od . [y,z])] = od . [(om . [x,y]),(om . [x,z])] & om . [(od . [x,y]),z] = od . [(om . [x,z]),(om . [y,z])] ) ) );
end;

:: deftheorem Def8 defines Field-like REALSET2:def 8 :
for IT being doubleLoopStr holds
( IT is Field-like iff ex A being non trivial set ex od being BinOp of A ex nd being Element of A ex om being DnT of nd,A ex nm being Element of A \ {nd} st
( IT = field A,od,om,nd,nm & LoopStr(# A,od,nd #) is Group & ( for B being non empty set
for P being BinOp of B
for e being Element of B st B = A \ {nd} & e = nm & P = om ! A,nd holds
LoopStr(# B,P,e #) is Group ) & ( for x, y, z being Element of A holds
( om . [x,(od . [y,z])] = od . [(om . [x,y]),(om . [x,z])] & om . [(od . [x,y]),z] = od . [(om . [x,z]),(om . [y,z])] ) ) ) );

registration
cluster strict Field-like doubleLoopStr ;
existence
ex b1 being doubleLoopStr st
( b1 is strict & b1 is Field-like )
proof end;
end;

definition
mode Field is Field-like doubleLoopStr ;
end;

definition
let F be Field;
func suppf c1 -> non trivial set means :Def9: :: REALSET2:def 9
ex od being BinOp of it ex nd being Element of it ex om being DnT of nd,it ex nm being Element of it \ {nd} st x = field it,od,om,nd,nm;
existence
ex b1 being non trivial set ex od being BinOp of b1 ex nd being Element of b1 ex om being DnT of nd,b1 ex nm being Element of b1 \ {nd} st F = field b1,od,om,nd,nm
proof end;
uniqueness
for b1, b2 being non trivial set st ex od being BinOp of b1 ex nd being Element of b1 ex om being DnT of nd,b1 ex nm being Element of b1 \ {nd} st F = field b1,od,om,nd,nm & ex od being BinOp of b2 ex nd being Element of b2 ex om being DnT of nd,b2 ex nm being Element of b2 \ {nd} st F = field b2,od,om,nd,nm holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines suppf REALSET2:def 9 :
for F being Field
for b2 being non trivial set holds
( b2 = suppf F iff ex od being BinOp of b2 ex nd being Element of b2 ex om being DnT of nd,b2 ex nm being Element of b2 \ {nd} st F = field b2,od,om,nd,nm );

definition
let F be Field;
func odf c1 -> BinOp of suppf a1 means :Def10: :: REALSET2:def 10
ex nd being Element of suppf x ex om being DnT of nd, suppf x ex nm being Element of (suppf x) \ {nd} st x = field (suppf x),it,om,nd,nm;
existence
ex b1 being BinOp of suppf F ex nd being Element of suppf F ex om being DnT of nd, suppf F ex nm being Element of (suppf F) \ {nd} st F = field (suppf F),b1,om,nd,nm
by ;
uniqueness
for b1, b2 being BinOp of suppf F st ex nd being Element of suppf F ex om being DnT of nd, suppf F ex nm being Element of (suppf F) \ {nd} st F = field (suppf F),b1,om,nd,nm & ex nd being Element of suppf F ex om being DnT of nd, suppf F ex nm being Element of (suppf F) \ {nd} st F = field (suppf F),b2,om,nd,nm holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines odf REALSET2:def 10 :
for F being Field
for b2 being BinOp of suppf F holds
( b2 = odf F iff ex nd being Element of suppf F ex om being DnT of nd, suppf F ex nm being Element of (suppf F) \ {nd} st F = field (suppf F),b2,om,nd,nm );

definition
let F be Field;
func ndf c1 -> Element of suppf a1 means :Def11: :: REALSET2:def 11
ex om being DnT of it, suppf x ex nm being Element of (suppf x) \ {it} st x = field (suppf x),(odf x),om,it,nm;
existence
ex b1 being Element of suppf F ex om being DnT of b1, suppf F ex nm being Element of (suppf F) \ {b1} st F = field (suppf F),(odf F),om,b1,nm
by Lemma51;
uniqueness
for b1, b2 being Element of suppf F st ex om being DnT of b1, suppf F ex nm being Element of (suppf F) \ {b1} st F = field (suppf F),(odf F),om,b1,nm & ex om being DnT of b2, suppf F ex nm being Element of (suppf F) \ {b2} st F = field (suppf F),(odf F),om,b2,nm holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines ndf REALSET2:def 11 :
for F being Field
for b2 being Element of suppf F holds
( b2 = ndf F iff ex om being DnT of b2, suppf F ex nm being Element of (suppf F) \ {b2} st F = field (suppf F),(odf F),om,b2,nm );

definition
let F be Field;
func omf c1 -> DnT of ndf a1, suppf a1 means :Def12: :: REALSET2:def 12
ex nm being Element of (suppf x) \ {(ndf x)} st x = field (suppf x),(odf x),it,(ndf x),nm;
existence
ex b1 being DnT of ndf F, suppf F ex nm being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),b1,(ndf F),nm
by Lemma52;
uniqueness
for b1, b2 being DnT of ndf F, suppf F st ex nm being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),b1,(ndf F),nm & ex nm being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),b2,(ndf F),nm holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines omf REALSET2:def 12 :
for F being Field
for b2 being DnT of ndf F, suppf F holds
( b2 = omf F iff ex nm being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),b2,(ndf F),nm );

definition
let F be Field;
func nmf c1 -> Element of (suppf a1) \ {(ndf a1)} means :Def13: :: REALSET2:def 13
x = field (suppf x),(odf x),(omf x),(ndf x),it;
existence
ex b1 being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),(omf F),(ndf F),b1
by ;
uniqueness
for b1, b2 being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),(omf F),(ndf F),b1 & F = field (suppf F),(odf F),(omf F),(ndf F),b2 holds
b1 = b2
proof end;
end;

:: deftheorem Def13 defines nmf REALSET2:def 13 :
for F being Field
for b2 being Element of (suppf F) \ {(ndf F)} holds
( b2 = nmf F iff F = field (suppf F),(odf F),(omf F),(ndf F),b2 );

theorem Th1: :: REALSET2:1
for F being Field holds LoopStr(# (suppf F),(odf F),(ndf F) #) is Group
proof end;

theorem Th2: :: REALSET2:2
for F being Field
for B being non empty set
for P being BinOp of B
for e being Element of B st B = (suppf F) \ {(ndf F)} & e = nmf F & P = (omf F) ! (suppf F),(ndf F) holds
LoopStr(# B,P,e #) is Group
proof end;

theorem Th3: :: REALSET2:3
for F being Field
for x, y, z being Element of suppf F holds
( (omf F) . [x,((odf F) . [y,z])] = (odf F) . [((omf F) . [x,y]),((omf F) . [x,z])] & (omf F) . [((odf F) . [x,y]),z] = (odf F) . [((omf F) . [x,z]),((omf F) . [y,z])] )
proof end;

theorem Th4: :: REALSET2:4
for F being Field
for a, b, c being Element of suppf F holds (odf F) . ((odf F) . a,b),c = (odf F) . a,((odf F) . b,c)
proof end;

theorem Th5: :: REALSET2:5
for F being Field
for a, b being Element of suppf F holds (odf F) . a,b = (odf F) . b,a
proof end;

theorem Th6: :: REALSET2:6
for F being Field
for a being Element of suppf F holds
( (odf F) . a,(ndf F) = a & (odf F) . (ndf F),a = a )
proof end;

theorem Th7: :: REALSET2:7
for F being Field
for a being Element of suppf F ex b being Element of suppf F st
( (odf F) . a,b = ndf F & (odf F) . b,a = ndf F )
proof end;

theorem Th8: :: REALSET2:8
for F being Field
for a, b, c being Element of (suppf F) \ {(ndf F)} holds (omf F) . [((omf F) . [a,b]),c] = (omf F) . [a,((omf F) . [b,c])]
proof end;

theorem Th9: :: REALSET2:9
for F being Field
for a, b being Element of (suppf F) \ {(ndf F)} holds (omf F) . [a,b] = (omf F) . [b,a]
proof end;

theorem Th10: :: REALSET2:10
for F being Field
for a being Element of (suppf F) \ {(ndf F)} holds
( (omf F) . [a,(nmf F)] = a & (omf F) . [(nmf F),a] = a )
proof end;

theorem Th11: :: REALSET2:11
for F being Field
for a being Element of (suppf F) \ {(ndf F)} ex b being Element of (suppf F) \ {(ndf F)} st
( (omf F) . [a,b] = nmf F & (omf F) . [b,a] = nmf F )
proof end;

definition
let F be Field;
func compf c1 -> Function of suppf a1, suppf a1 means :Def14: :: REALSET2:def 14
for x being Element of suppf x holds (odf x) . [x,(it . x)] = ndf x;
existence
ex b1 being Function of suppf F, suppf F st
for x being Element of suppf F holds (odf F) . [x,(b1 . x)] = ndf F
proof end;
uniqueness
for b1, b2 being Function of suppf F, suppf F st ( for x being Element of suppf F holds (odf F) . [x,(b1 . x)] = ndf F ) & ( for x being Element of suppf F holds (odf F) . [x,(b2 . x)] = ndf F ) holds
b1 = b2
proof end;
end;

:: deftheorem Def14 defines compf REALSET2:def 14 :
for F being Field
for b2 being Function of suppf F, suppf F holds
( b2 = compf F iff for x being Element of suppf F holds (odf F) . [x,(b2 . x)] = ndf F );

theorem Th12: :: REALSET2:12
for F being Field
for x, y being Element of suppf F st (odf F) . x,y = ndf F holds
y = (compf F) . x
proof end;

theorem Th13: :: REALSET2:13
for F being Field
for x being Element of suppf F holds x = (compf F) . ((compf F) . x)
proof end;

theorem Th14: :: REALSET2:14
for F being Field
for a, b being Element of suppf F holds
( (odf F) . [a,b] is Element of suppf F & (omf F) . [a,b] is Element of suppf F & (compf F) . a is Element of suppf F )
proof end;

theorem Th15: :: REALSET2:15
for F being Field
for a, b, c being Element of suppf F holds (omf F) . [a,((odf F) . [b,((compf F) . c)])] = (odf F) . [((omf F) . [a,b]),((compf F) . ((omf F) . [a,c]))]
proof end;

theorem Th16: :: REALSET2:16
for F being Field
for a, b, c being Element of suppf F holds (omf F) . [((odf F) . [a,((compf F) . b)]),c] = (odf F) . [((omf F) . [a,c]),((compf F) . ((omf F) . [b,c]))]
proof end;

theorem Th17: :: REALSET2:17
for F being Field
for a being Element of suppf F holds (omf F) . [a,(ndf F)] = ndf F
proof end;

theorem Th18: :: REALSET2:18
for F being Field
for a being Element of suppf F holds (omf F) . [(ndf F),a] = ndf F
proof end;

theorem Th19: :: REALSET2:19
for F being Field
for a, b being Element of suppf F holds (compf F) . ((omf F) . [a,b]) = (omf F) . [a,((compf F) . b)]
proof end;

theorem Th20: :: REALSET2:20
for F being Field holds (omf F) . [(nmf F),(ndf F)] = ndf F by Lemma71;

theorem Th21: :: REALSET2:21
for F being Field holds (omf F) . [(ndf F),(nmf F)] = ndf F by Lemma72;

theorem Th22: :: REALSET2:22
for F being Field
for a, b being Element of suppf F holds (omf F) . [a,b] is Element of suppf F by ;

theorem Th23: :: REALSET2:23
for F being Field
for a, b, c being Element of suppf F holds (omf F) . [((omf F) . [a,b]),c] = (omf F) . [a,((omf F) . [b,c])]
proof end;

theorem Th24: :: REALSET2:24
for F being Field
for a, b being Element of suppf F holds (omf F) . [a,b] = (omf F) . [b,a]
proof end;

theorem Th25: :: REALSET2:25
for F being Field
for a being Element of suppf F holds
( (omf F) . [a,(nmf F)] = a & (omf F) . [(nmf F),a] = a )
proof end;

definition
let F be Field;
func revf c1 -> Function of (suppf a1) \ {(ndf a1)},(suppf a1) \ {(ndf a1)} means :Def15: :: REALSET2:def 15
for x being Element of (suppf x) \ {(ndf x)} holds (omf x) . [x,(it . x)] = nmf x;
existence
ex b1 being Function of (suppf F) \ {(ndf F)},(suppf F) \ {(ndf F)} st
for x being Element of (suppf F) \ {(ndf F)} holds (omf F) . [x,(b1 . x)] = nmf F
proof end;
uniqueness
for b1, b2 being Function of (suppf F) \ {(ndf F)},(suppf F) \ {(ndf F)} st ( for x being Element of (suppf F) \ {(ndf F)} holds (omf F) . [x,(b1 . x)] = nmf F ) & ( for x being Element of (suppf F) \ {(ndf F)} holds (omf F) . [x,(b2 . x)] = nmf F ) holds
b1 = b2
proof end;
end;

:: deftheorem Def15 defines revf REALSET2:def 15 :
for F being Field
for b2 being Function of (suppf F) \ {(ndf F)},(suppf F) \ {(ndf F)} holds
( b2 = revf F iff for x being Element of (suppf F) \ {(ndf F)} holds (omf F) . [x,(b2 . x)] = nmf F );

theorem Th26: :: REALSET2:26
for F being Field
for x, y being Element of (suppf F) \ {(ndf F)} st (omf F) . [x,y] = nmf F holds
y = (revf F) . x
proof end;

theorem Th27: :: REALSET2:27
for F being Field
for x being Element of (suppf F) \ {(ndf F)} holds x = (revf F) . ((revf F) . x)
proof end;

theorem Th28: :: REALSET2:28
for F being Field
for a, b being Element of (suppf F) \ {(ndf F)} holds
( (omf F) . [a,b] is Element of (suppf F) \ {(ndf F)} & (revf F) . a is Element of (suppf F) \ {(ndf F)} )
proof end;

theorem Th29: :: REALSET2:29
for F being Field
for a, b, c being Element of suppf F st (odf F) . a,b = (odf F) . a,c holds
b = c
proof end;

theorem Th30: :: REALSET2:30
for F being Field
for a being Element of (suppf F) \ {(ndf F)}
for b, c being Element of suppf F st (omf F) . [a,b] = (omf F) . [a,c] holds
b = c
proof end;