:: VALUED_0 semantic presentation

definition
let f be Relation;
attr f is complex-valued means :Def1: :: VALUED_0:def 1
rng f c= COMPLEX ;
attr f is ext-real-valued means :Def2: :: VALUED_0:def 2
rng f c= ExtREAL ;
attr f is real-valued means :Def3: :: VALUED_0:def 3
rng f c= REAL ;
attr f is rational-valued means :Def4: :: VALUED_0:def 4
rng f c= RAT ;
attr f is integer-valued means :Def5: :: VALUED_0:def 5
rng f c= INT ;
attr f is natural-valued means :Def6: :: VALUED_0:def 6
rng f c= NAT ;
end;

:: deftheorem Def1 defines complex-valued VALUED_0:def 1 :
for f being Relation holds
( f is complex-valued iff rng f c= COMPLEX );

:: deftheorem Def2 defines ext-real-valued VALUED_0:def 2 :
for f being Relation holds
( f is ext-real-valued iff rng f c= ExtREAL );

:: deftheorem Def3 defines real-valued VALUED_0:def 3 :
for f being Relation holds
( f is real-valued iff rng f c= REAL );

:: deftheorem Def4 defines rational-valued VALUED_0:def 4 :
for f being Relation holds
( f is rational-valued iff rng f c= RAT );

:: deftheorem Def5 defines integer-valued VALUED_0:def 5 :
for f being Relation holds
( f is integer-valued iff rng f c= INT );

:: deftheorem Def6 defines natural-valued VALUED_0:def 6 :
for f being Relation holds
( f is natural-valued iff rng f c= NAT );

registration
cluster natural-valued -> integer-valued set ;
coherence
for b1 being Relation st b1 is natural-valued holds
b1 is integer-valued
proof end;
cluster integer-valued -> rational-valued set ;
coherence
for b1 being Relation st b1 is integer-valued holds
b1 is rational-valued
proof end;
cluster rational-valued -> real-valued set ;
coherence
for b1 being Relation st b1 is rational-valued holds
b1 is real-valued
proof end;
cluster real-valued -> ext-real-valued set ;
coherence
for b1 being Relation st b1 is real-valued holds
b1 is ext-real-valued
proof end;
cluster real-valued -> complex-valued set ;
coherence
for b1 being Relation st b1 is real-valued holds
b1 is complex-valued
proof end;
end;

registration
cluster empty -> natural-valued set ;
coherence
for b1 being Relation st b1 is empty holds
b1 is natural-valued
proof end;
end;

registration
cluster {} -> natural-valued ;
coherence
{} is natural-valued
;
end;

registration
cluster natural-valued set ;
existence
ex b1 being Function st b1 is natural-valued
proof end;
end;

registration
let R be complex-valued Relation;
cluster rng R -> complex-membered ;
coherence
rng R is complex-membered
proof end;
end;

registration
let R be ext-real-valued Relation;
cluster rng R -> ext-real-membered ;
coherence
rng R is ext-real-membered
proof end;
end;

registration
let R be real-valued Relation;
cluster rng R -> real-membered ;
coherence
rng R is real-membered
proof end;
end;

registration
let R be rational-valued Relation;
cluster rng R -> rational-membered ;
coherence
rng R is rational-membered
proof end;
end;

registration
let R be integer-valued Relation;
cluster rng R -> integer-membered ;
coherence
rng R is integer-membered
proof end;
end;

registration
let R be natural-valued Relation;
cluster rng R -> natural-membered ;
coherence
rng R is natural-membered
proof end;
end;

theorem Th1: :: VALUED_0:1
for R being Relation
for S being complex-valued Relation st R c= S holds
R is complex-valued
proof end;

theorem Th2: :: VALUED_0:2
for R being Relation
for S being ext-real-valued Relation st R c= S holds
R is ext-real-valued
proof end;

theorem Th3: :: VALUED_0:3
for R being Relation
for S being real-valued Relation st R c= S holds
R is real-valued
proof end;

theorem Th4: :: VALUED_0:4
for R being Relation
for S being rational-valued Relation st R c= S holds
R is rational-valued
proof end;

theorem Th5: :: VALUED_0:5
for R being Relation
for S being integer-valued Relation st R c= S holds
R is integer-valued
proof end;

theorem Th6: :: VALUED_0:6
for R being Relation
for S being natural-valued Relation st R c= S holds
R is natural-valued
proof end;

registration
let R be complex-valued Relation;
cluster -> complex-valued Element of bool R;
coherence
for b1 being Subset of R holds b1 is complex-valued
by Th1;
end;

registration
let R be ext-real-valued Relation;
cluster -> ext-real-valued Element of bool R;
coherence
for b1 being Subset of R holds b1 is ext-real-valued
by Th2;
end;

registration
let R be real-valued Relation;
cluster -> real-valued Element of bool R;
coherence
for b1 being Subset of R holds b1 is real-valued
by Th3;
end;

registration
let R be rational-valued Relation;
cluster -> rational-valued Element of bool R;
coherence
for b1 being Subset of R holds b1 is rational-valued
by Th4;
end;

registration
let R be integer-valued Relation;
cluster -> integer-valued Element of bool R;
coherence
for b1 being Subset of R holds b1 is integer-valued
by Th5;
end;

registration
let R be natural-valued Relation;
cluster -> natural-valued Element of bool R;
coherence
for b1 being Subset of R holds b1 is natural-valued
by Th6;
end;

registration
let R, S be complex-valued Relation;
cluster R \/ S -> complex-valued ;
coherence
R \/ S is complex-valued
proof end;
end;

registration
let R, S be ext-real-valued Relation;
cluster R \/ S -> ext-real-valued ;
coherence
R \/ S is ext-real-valued
proof end;
end;

registration
let R, S be real-valued Relation;
cluster R \/ S -> real-valued ;
coherence
R \/ S is real-valued
proof end;
end;

registration
let R, S be rational-valued Relation;
cluster R \/ S -> rational-valued ;
coherence
R \/ S is rational-valued
proof end;
end;

registration
let R, S be integer-valued Relation;
cluster R \/ S -> integer-valued ;
coherence
R \/ S is integer-valued
proof end;
end;

registration
let R, S be natural-valued Relation;
cluster R \/ S -> natural-valued ;
coherence
R \/ S is natural-valued
proof end;
end;

registration
let R be complex-valued Relation;
let S be Relation;
cluster R /\ S -> complex-valued ;
coherence
R /\ S is complex-valued
proof end;
cluster R \ S -> complex-valued ;
coherence
R \ S is complex-valued
;
end;

registration
let R be ext-real-valued Relation;
let S be Relation;
cluster R /\ S -> ext-real-valued ;
coherence
R /\ S is ext-real-valued
proof end;
cluster R \ S -> ext-real-valued ;
coherence
R \ S is ext-real-valued
;
end;

registration
let R be real-valued Relation;
let S be Relation;
cluster R /\ S -> real-valued ;
coherence
R /\ S is real-valued
proof end;
cluster R \ S -> real-valued ;
coherence
R \ S is real-valued
;
end;

registration
let R be rational-valued Relation;
let S be Relation;
cluster R /\ S -> rational-valued ;
coherence
R /\ S is rational-valued
proof end;
cluster R \ S -> rational-valued ;
coherence
R \ S is rational-valued
;
end;

registration
let R be integer-valued Relation;
let S be Relation;
cluster R /\ S -> integer-valued ;
coherence
R /\ S is integer-valued
proof end;
cluster R \ S -> integer-valued ;
coherence
R \ S is integer-valued
;
end;

registration
let R be natural-valued Relation;
let S be Relation;
cluster R /\ S -> natural-valued ;
coherence
R /\ S is natural-valued
proof end;
cluster R \ S -> natural-valued ;
coherence
R \ S is natural-valued
;
end;

registration
let R, S be complex-valued Relation;
cluster R \+\ S -> complex-valued ;
coherence
R \+\ S is complex-valued
;
end;

registration
let R, S be ext-real-valued Relation;
cluster R \+\ S -> ext-real-valued ;
coherence
R \+\ S is ext-real-valued
;
end;

registration
let R, S be real-valued Relation;
cluster R \+\ S -> real-valued ;
coherence
R \+\ S is real-valued
;
end;

registration
let R, S be rational-valued Relation;
cluster R \+\ S -> rational-valued ;
coherence
R \+\ S is rational-valued
;
end;

registration
let R, S be integer-valued Relation;
cluster R \+\ S -> integer-valued ;
coherence
R \+\ S is integer-valued
;
end;

registration
let R, S be natural-valued Relation;
cluster R \+\ S -> natural-valued ;
coherence
R \+\ S is natural-valued
;
end;

registration
let R be complex-valued Relation;
let X be set ;
cluster R .: X -> complex-membered ;
coherence
R .: X is complex-membered
proof end;
end;

registration
let R be ext-real-valued Relation;
let X be set ;
cluster R .: X -> ext-real-membered ;
coherence
R .: X is ext-real-membered
proof end;
end;

registration
let R be real-valued Relation;
let X be set ;
cluster R .: X -> real-membered ;
coherence
R .: X is real-membered
proof end;
end;

registration
let R be rational-valued Relation;
let X be set ;
cluster R .: X -> rational-membered ;
coherence
R .: X is rational-membered
proof end;
end;

registration
let R be integer-valued Relation;
let X be set ;
cluster R .: X -> integer-membered ;
coherence
R .: X is integer-membered
proof end;
end;

registration
let R be natural-valued Relation;
let X be set ;
cluster R .: X -> natural-membered ;
coherence
R .: X is natural-membered
proof end;
end;

registration
let R be complex-valued Relation;
let x be set ;
cluster Im R,x -> complex-membered ;
coherence
Im R,x is complex-membered
;
end;

registration
let R be ext-real-valued Relation;
let x be set ;
cluster Im R,x -> ext-real-membered ;
coherence
Im R,x is ext-real-membered
;
end;

registration
let R be real-valued Relation;
let x be set ;
cluster Im R,x -> real-membered ;
coherence
Im R,x is real-membered
;
end;

registration
let R be rational-valued Relation;
let x be set ;
cluster Im R,x -> rational-membered ;
coherence
Im R,x is rational-membered
;
end;

registration
let R be integer-valued Relation;
let x be set ;
cluster Im R,x -> integer-membered ;
coherence
Im R,x is integer-membered
;
end;

registration
let R be natural-valued Relation;
let x be set ;
cluster Im R,x -> natural-membered ;
coherence
Im R,x is natural-membered
;
end;

registration
let R be complex-valued Relation;
let X be set ;
cluster R | X -> complex-valued ;
coherence
R | X is complex-valued
proof end;
end;

registration
let R be ext-real-valued Relation;
let X be set ;
cluster R | X -> ext-real-valued ;
coherence
R | X is ext-real-valued
proof end;
end;

registration
let R be real-valued Relation;
let X be set ;
cluster R | X -> real-valued ;
coherence
R | X is real-valued
proof end;
end;

registration
let R be rational-valued Relation;
let X be set ;
cluster R | X -> rational-valued ;
coherence
R | X is rational-valued
proof end;
end;

registration
let R be integer-valued Relation;
let X be set ;
cluster R | X -> integer-valued ;
coherence
R | X is integer-valued
proof end;
end;

registration
let R be natural-valued Relation;
let X be set ;
cluster R | X -> natural-valued ;
coherence
R | X is natural-valued
proof end;
end;

registration
let X be complex-membered set ;
cluster id X -> complex-valued ;
coherence
id X is complex-valued
proof end;
end;

registration
let X be ext-real-membered set ;
cluster id X -> ext-real-valued ;
coherence
id X is ext-real-valued
proof end;
end;

registration
let X be real-membered set ;
cluster id X -> real-valued ;
coherence
id X is real-valued
proof end;
end;

registration
let X be rational-membered set ;
cluster id X -> rational-valued ;
coherence
id X is rational-valued
proof end;
end;

registration
let X be integer-membered set ;
cluster id X -> integer-valued ;
coherence
id X is integer-valued
proof end;
end;

registration
let X be natural-membered set ;
cluster id X -> natural-valued ;
coherence
id X is natural-valued
proof end;
end;

registration
let R be Relation;
let S be complex-valued Relation;
cluster R * S -> complex-valued ;
coherence
R * S is complex-valued
proof end;
end;

registration
let R be Relation;
let S be ext-real-valued Relation;
cluster R * S -> ext-real-valued ;
coherence
R * S is ext-real-valued
proof end;
end;

registration
let R be Relation;
let S be real-valued Relation;
cluster R * S -> real-valued ;
coherence
R * S is real-valued
proof end;
end;

registration
let R be Relation;
let S be rational-valued Relation;
cluster R * S -> rational-valued ;
coherence
R * S is rational-valued
proof end;
end;

registration
let R be Relation;
let S be integer-valued Relation;
cluster R * S -> integer-valued ;
coherence
R * S is integer-valued
proof end;
end;

registration
let R be Relation;
let S be natural-valued Relation;
cluster R * S -> natural-valued ;
coherence
R * S is natural-valued
proof end;
end;

definition
let f be Function;
redefine attr f is complex-valued means :Def7: :: VALUED_0:def 7
for x being set st x in dom f holds
f . x is complex;
compatibility
( f is complex-valued iff for x being set st x in dom f holds
f . x is complex )
proof end;
redefine attr f is ext-real-valued means :Def8: :: VALUED_0:def 8
for x being set st x in dom f holds
f . x is ext-real;
compatibility
( f is ext-real-valued iff for x being set st x in dom f holds
f . x is ext-real )
proof end;
redefine attr f is real-valued means :Def9: :: VALUED_0:def 9
for x being set st x in dom f holds
f . x is real;
compatibility
( f is real-valued iff for x being set st x in dom f holds
f . x is real )
proof end;
redefine attr f is rational-valued means :Def10: :: VALUED_0:def 10
for x being set st x in dom f holds
f . x is rational;
compatibility
( f is rational-valued iff for x being set st x in dom f holds
f . x is rational )
proof end;
redefine attr f is integer-valued means :Def11: :: VALUED_0:def 11
for x being set st x in dom f holds
f . x is integer;
compatibility
( f is integer-valued iff for x being set st x in dom f holds
f . x is integer )
proof end;
redefine attr f is natural-valued means :Def12: :: VALUED_0:def 12
for x being set st x in dom f holds
f . x is natural;
compatibility
( f is natural-valued iff for x being set st x in dom f holds
f . x is natural )
proof end;
end;

:: deftheorem Def7 defines complex-valued VALUED_0:def 7 :
for f being Function holds
( f is complex-valued iff for x being set st x in dom f holds
f . x is complex );

:: deftheorem Def8 defines ext-real-valued VALUED_0:def 8 :
for f being Function holds
( f is ext-real-valued iff for x being set st x in dom f holds
f . x is ext-real );

:: deftheorem Def9 defines real-valued VALUED_0:def 9 :
for f being Function holds
( f is real-valued iff for x being set st x in dom f holds
f . x is real );

:: deftheorem Def10 defines rational-valued VALUED_0:def 10 :
for f being Function holds
( f is rational-valued iff for x being set st x in dom f holds
f . x is rational );

:: deftheorem Def11 defines integer-valued VALUED_0:def 11 :
for f being Function holds
( f is integer-valued iff for x being set st x in dom f holds
f . x is integer );

:: deftheorem Def12 defines natural-valued VALUED_0:def 12 :
for f being Function holds
( f is natural-valued iff for x being set st x in dom f holds
f . x is natural );

theorem Th7: :: VALUED_0:7
for f being Function holds
( f is complex-valued iff for x being set holds f . x is complex )
proof end;

theorem Th8: :: VALUED_0:8
for f being Function holds
( f is ext-real-valued iff for x being set holds f . x is ext-real )
proof end;

theorem Th9: :: VALUED_0:9
for f being Function holds
( f is real-valued iff for x being set holds f . x is real )
proof end;

theorem Th10: :: VALUED_0:10
for f being Function holds
( f is rational-valued iff for x being set holds f . x is rational )
proof end;

theorem Th11: :: VALUED_0:11
for f being Function holds
( f is integer-valued iff for x being set holds f . x is integer )
proof end;

theorem Th12: :: VALUED_0:12
for f being Function holds
( f is natural-valued iff for x being set holds f . x is natural )
proof end;

registration
let f be complex-valued Function;
let x be set ;
cluster f . x -> complex ;
coherence
f . x is complex
by Th7;
end;

registration
let f be ext-real-valued Function;
let x be set ;
cluster f . x -> ext-real ;
coherence
f . x is ext-real
by Th8;
end;

registration
let f be real-valued Function;
let x be set ;
cluster f . x -> real ;
coherence
f . x is real
by Th9;
end;

registration
let f be rational-valued Function;
let x be set ;
cluster f . x -> rational ;
coherence
f . x is rational
by Th10;
end;

registration
let f be integer-valued Function;
let x be set ;
cluster f . x -> integer ;
coherence
f . x is integer
by Th11;
end;

registration
let f be natural-valued Function;
let x be set ;
cluster f . x -> natural ;
coherence
f . x is natural
by Th12;
end;

registration
let X be set ;
let x be complex number ;
cluster X --> x -> complex-valued ;
coherence
X --> x is complex-valued
proof end;
end;

registration
let X be set ;
let x be ext-real number ;
cluster X --> x -> ext-real-valued ;
coherence
X --> x is ext-real-valued
proof end;
end;

registration
let X be set ;
let x be real number ;
cluster X --> x -> real-valued ;
coherence
X --> x is real-valued
proof end;
end;

registration
let X be set ;
let x be rational number ;
cluster X --> x -> rational-valued ;
coherence
X --> x is rational-valued
proof end;
end;

registration
let X be set ;
let x be integer number ;
cluster X --> x -> integer-valued ;
coherence
X --> x is integer-valued
proof end;
end;

registration
let X be set ;
let x be natural number ;
cluster X --> x -> natural-valued ;
coherence
X --> x is natural-valued
proof end;
end;

registration
let f, g be complex-valued Function;
cluster f +* g -> complex-valued ;
coherence
f +* g is complex-valued
proof end;
end;

registration
let f, g be ext-real-valued Function;
cluster f +* g -> ext-real-valued ;
coherence
f +* g is ext-real-valued
proof end;
end;

registration
let f, g be real-valued Function;
cluster f +* g -> real-valued ;
coherence
f +* g is real-valued
proof end;
end;

registration
let f, g be rational-valued Function;
cluster f +* g -> rational-valued ;
coherence
f +* g is rational-valued
proof end;
end;

registration
let f, g be integer-valued Function;
cluster f +* g -> integer-valued ;
coherence
f +* g is integer-valued
proof end;
end;

registration
let f, g be natural-valued Function;
cluster f +* g -> natural-valued ;
coherence
f +* g is natural-valued
proof end;
end;

registration
let x be set ;
let y be complex number ;
cluster x .--> y -> complex-valued ;
coherence
x .--> y is complex-valued
;
end;

registration
let x be set ;
let y be ext-real number ;
cluster x .--> y -> ext-real-valued ;
coherence
x .--> y is ext-real-valued
;
end;

registration
let x be set ;
let y be real number ;
cluster x .--> y -> real-valued ;
coherence
x .--> y is real-valued
;
end;

registration
let x be set ;
let y be rational number ;
cluster x .--> y -> rational-valued ;
coherence
x .--> y is rational-valued
;
end;

registration
let x be set ;
let y be integer number ;
cluster x .--> y -> integer-valued ;
coherence
x .--> y is integer-valued
;
end;

registration
let x be set ;
let y be natural number ;
cluster x .--> y -> natural-valued ;
coherence
x .--> y is natural-valued
;
end;

registration
let X be set ;
let Y be complex-membered set ;
cluster -> complex-valued Relation of X,Y;
coherence
for b1 being Relation of X,Y holds b1 is complex-valued
proof end;
end;

registration
let X be set ;
let Y be ext-real-membered set ;
cluster -> ext-real-valued Relation of X,Y;
coherence
for b1 being Relation of X,Y holds b1 is ext-real-valued
proof end;
end;

registration
let X be set ;
let Y be real-membered set ;
cluster -> real-valued Relation of X,Y;
coherence
for b1 being Relation of X,Y holds b1 is real-valued
proof end;
end;

registration
let X be set ;
let Y be rational-membered set ;
cluster -> rational-valued Relation of X,Y;
coherence
for b1 being Relation of X,Y holds b1 is rational-valued
proof end;
end;

registration
let X be set ;
let Y be integer-membered set ;
cluster -> integer-valued Relation of X,Y;
coherence
for b1 being Relation of X,Y holds b1 is integer-valued
proof end;
end;

registration
let X be set ;
let Y be natural-membered set ;
cluster -> natural-valued Relation of X,Y;
coherence
for b1 being Relation of X,Y holds b1 is natural-valued
proof end;
end;

registration
let X be set ;
let Y be complex-membered set ;
cluster [:X,Y:] -> complex-valued ;
coherence
[:X,Y:] is complex-valued
proof end;
end;

registration
let X be set ;
let Y be ext-real-membered set ;
cluster [:X,Y:] -> ext-real-valued ;
coherence
[:X,Y:] is ext-real-valued
proof end;
end;

registration
let X be set ;
let Y be real-membered set ;
cluster [:X,Y:] -> real-valued ;
coherence
[:X,Y:] is real-valued
proof end;
end;

registration
let X be set ;
let Y be rational-membered set ;
cluster [:X,Y:] -> rational-valued ;
coherence
[:X,Y:] is rational-valued
proof end;
end;

registration
let X be set ;
let Y be integer-membered set ;
cluster [:X,Y:] -> integer-valued ;
coherence
[:X,Y:] is integer-valued
proof end;
end;

registration
let X be set ;
let Y be natural-membered set ;
cluster [:X,Y:] -> natural-valued ;
coherence
[:X,Y:] is natural-valued
proof end;
end;