:: MEASURE6 semantic presentation

theorem :: MEASURE6:1
ex F being Function of NAT ,[:NAT ,NAT :] st
( F is one-to-one & dom F = NAT & rng F = [:NAT ,NAT :] )
proof end;

theorem :: MEASURE6:2
for F being Function of NAT , ExtREAL st F is nonnegative holds
0. <= SUM F
proof end;

theorem :: MEASURE6:3
for F being Function of NAT , ExtREAL
for x being R_eal st ex n being Element of NAT st x <= F . n & F is nonnegative holds
x <= SUM F
proof end;

theorem :: MEASURE6:4
canceled;

theorem :: MEASURE6:5
canceled;

theorem :: MEASURE6:6
canceled;

theorem :: MEASURE6:7
canceled;

theorem Th8: :: MEASURE6:8
for x, y being R_eal st x is Real holds
( (y - x) + x = y & (y + x) - x = y )
proof end;

theorem :: MEASURE6:9
canceled;

theorem Th10: :: MEASURE6:10
for x, y, z being R_eal st z in REAL & y < x holds
(z + x) - (z + y) = x - y
proof end;

theorem Th11: :: MEASURE6:11
for x, y, z being R_eal st z in REAL & x <= y holds
( z + x <= z + y & x + z <= y + z & x - z <= y - z )
proof end;

theorem Th12: :: MEASURE6:12
for x, y, z being R_eal st z in REAL & x < y holds
( z + x < z + y & x + z < y + z & x - z < y - z )
proof end;

definition
let x be real number ;
func R_EAL x -> R_eal equals :: MEASURE6:def 1
x;
coherence
x is R_eal
proof end;
end;

:: deftheorem defines R_EAL MEASURE6:def 1 :
for x being real number holds R_EAL x = x;

theorem :: MEASURE6:13
for x, y being real number holds
( x <= y iff R_EAL x <= R_EAL y ) ;

theorem :: MEASURE6:14
for x, y being real number holds
( x < y iff R_EAL x < R_EAL y ) ;

theorem Th15: :: MEASURE6:15
for x, y, z being R_eal st x < y & y < z holds
y is Real
proof end;

theorem :: MEASURE6:16
for x, y, z being R_eal st x is Real & z is Real & x <= y & y <= z holds
y is Real
proof end;

theorem Th17: :: MEASURE6:17
for x, y, z being R_eal st x is Real & x <= y & y < z holds
y is Real
proof end;

theorem Th18: :: MEASURE6:18
for x, y, z being R_eal st x < y & y <= z & z is Real holds
y is Real
proof end;

theorem Th19: :: MEASURE6:19
for x, y being R_eal st 0. < x & x < y holds
0. < y - x
proof end;

theorem :: MEASURE6:20
for x, y, z being R_eal st 0. <= x & 0. <= z & z + x < y holds
z < y - x
proof end;

theorem Th21: :: MEASURE6:21
for x being R_eal holds x - 0. = x
proof end;

theorem Th22: :: MEASURE6:22
for x, y, z being R_eal st 0. <= x & 0. <= z & z + x < y holds
z <= y
proof end;

theorem Th23: :: MEASURE6:23
for x being R_eal st 0. < x holds
ex y being R_eal st
( 0. < y & y < x )
proof end;

theorem Th24: :: MEASURE6:24
for x, z being R_eal st 0. < x & x < z holds
ex y being R_eal st
( 0. < y & x + y < z & y in REAL )
proof end;

theorem :: MEASURE6:25
for x, z being R_eal st 0. <= x & x < z holds
ex y being R_eal st
( 0. < y & x + y < z & y in REAL )
proof end;

theorem Th26: :: MEASURE6:26
for x being R_eal st 0. < x holds
ex y being R_eal st
( 0. < y & y + y < x )
proof end;

definition
let x be R_eal;
assume A1: 0. < x ;
func Seg x -> non empty Subset of ExtREAL means :Def2: :: MEASURE6:def 2
for y being R_eal holds
( y in it iff ( 0. < y & y + y < x ) );
existence
ex b1 being non empty Subset of ExtREAL st
for y being R_eal holds
( y in b1 iff ( 0. < y & y + y < x ) )
proof end;
uniqueness
for b1, b2 being non empty Subset of ExtREAL st ( for y being R_eal holds
( y in b1 iff ( 0. < y & y + y < x ) ) ) & ( for y being R_eal holds
( y in b2 iff ( 0. < y & y + y < x ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines Seg MEASURE6:def 2 :
for x being R_eal st 0. < x holds
for b2 being non empty Subset of ExtREAL holds
( b2 = Seg x iff for y being R_eal holds
( y in b2 iff ( 0. < y & y + y < x ) ) );

definition
let x be R_eal;
func len x -> R_eal equals :: MEASURE6:def 3
sup (Seg x);
correctness
coherence
sup (Seg x) is R_eal
;
;
end;

:: deftheorem defines len MEASURE6:def 3 :
for x being R_eal holds len x = sup (Seg x);

theorem Th27: :: MEASURE6:27
for x being R_eal st 0. < x holds
0. < len x
proof end;

theorem Th28: :: MEASURE6:28
for x being R_eal st 0. < x holds
len x <= x
proof end;

theorem Th29: :: MEASURE6:29
for x being R_eal st 0. < x & x < +infty holds
len x is Real
proof end;

theorem Th30: :: MEASURE6:30
for x being R_eal st 0. < x holds
(len x) + (len x) <= x
proof end;

theorem :: MEASURE6:31
for eps being R_eal st 0. < eps holds
ex F being Function of NAT , ExtREAL st
( ( for n being Element of NAT holds 0. < F . n ) & SUM F < eps )
proof end;

theorem :: MEASURE6:32
for eps being R_eal
for X being non empty Subset of ExtREAL st 0. < eps & inf X is Real holds
ex x being R_eal st
( x in X & x < (inf X) + eps )
proof end;

theorem :: MEASURE6:33
for eps being R_eal
for X being non empty Subset of ExtREAL st 0. < eps & sup X is Real holds
ex x being R_eal st
( x in X & (sup X) - eps < x )
proof end;

theorem :: MEASURE6:34
for F being Function of NAT , ExtREAL st F is nonnegative & SUM F < +infty holds
for n being Element of NAT holds F . n in REAL
proof end;

theorem :: MEASURE6:35
( REAL is Interval & REAL = ].-infty ,+infty .[ & REAL = [.-infty ,+infty .] & REAL = [.-infty ,+infty .[ & REAL = ].-infty ,+infty .] )
proof end;

theorem Th36: :: MEASURE6:36
for a, b being R_eal st b = -infty holds
( ].a,b.[ = {} & [.a,b.] = {} & [.a,b.[ = {} & ].a,b.] = {} )
proof end;

theorem Th37: :: MEASURE6:37
for a, b being R_eal st a = +infty holds
( ].a,b.[ = {} & [.a,b.] = {} & [.a,b.[ = {} & ].a,b.] = {} )
proof end;

theorem Th38: :: MEASURE6:38
for A being Interval
for a, b being R_eal
for c, d, e being Real st A = ].a,b.[ & c in A & d in A & c <= e & e <= d holds
e in A
proof end;

theorem Th39: :: MEASURE6:39
for A being Interval
for a, b being R_eal
for c, d, e being Real st A = [.a,b.] & c in A & d in A & c <= e & e <= d holds
e in A
proof end;

theorem Th40: :: MEASURE6:40
for A being Interval
for a, b being R_eal
for c, d, e being Real st A = ].a,b.] & c in A & d in A & c <= e & e <= d holds
e in A
proof end;

theorem Th41: :: MEASURE6:41
for A being Interval
for a, b being R_eal
for c, d, e being Real st A = [.a,b.[ & c in A & d in A & c <= e & e <= d holds
e in A
proof end;

theorem Th42: :: MEASURE6:42
for A being non empty Subset of ExtREAL
for m, M being R_eal st m = inf A & M = sup A & ( for c, d being Real st c in A & d in A holds
for e being Real st c <= e & e <= d holds
e in A ) & not m in A & not M in A holds
A = ].m,M.[
proof end;

theorem Th43: :: MEASURE6:43
for A being non empty Subset of ExtREAL
for m, M being R_eal st m = inf A & M = sup A & ( for c, d being Real st c in A & d in A holds
for e being Real st c <= e & e <= d holds
e in A ) & m in A & M in A & A c= REAL holds
A = [.m,M.]
proof end;

theorem Th44: :: MEASURE6:44
for A being non empty Subset of ExtREAL
for m, M being R_eal st m = inf A & M = sup A & ( for c, d being Real st c in A & d in A holds
for e being Real st c <= e & e <= d holds
e in A ) & m in A & not M in A & A c= REAL holds
A = [.m,M.[
proof end;

theorem Th45: :: MEASURE6:45
for A being non empty Subset of ExtREAL
for m, M being R_eal st m = inf A & M = sup A & ( for c, d being Real st c in A & d in A holds
for e being Real st c <= e & e <= d holds
e in A ) & not m in A & M in A & A c= REAL holds
A = ].m,M.]
proof end;

theorem Th46: :: MEASURE6:46
for A being Subset of REAL holds
( A is Interval iff for a, b being Real st a in A & b in A holds
for c being Real st a <= c & c <= b holds
c in A )
proof end;

theorem :: MEASURE6:47
for A, B being Interval st A meets B holds
A \/ B is Interval
proof end;

definition
let A be Interval;
assume A1: A <> {} ;
func ^^ A -> R_eal means :Def4: :: MEASURE6:def 4
ex b being R_eal st
( it <= b & ( A = ].it,b.[ or A = ].it,b.] or A = [.it,b.] or A = [.it,b.[ ) );
existence
ex b1, b being R_eal st
( b1 <= b & ( A = ].b1,b.[ or A = ].b1,b.] or A = [.b1,b.] or A = [.b1,b.[ ) )
proof end;
uniqueness
for b1, b2 being R_eal st ex b being R_eal st
( b1 <= b & ( A = ].b1,b.[ or A = ].b1,b.] or A = [.b1,b.] or A = [.b1,b.[ ) ) & ex b being R_eal st
( b2 <= b & ( A = ].b2,b.[ or A = ].b2,b.] or A = [.b2,b.] or A = [.b2,b.[ ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines ^^ MEASURE6:def 4 :
for A being Interval st A <> {} holds
for b2 being R_eal holds
( b2 = ^^ A iff ex b being R_eal st
( b2 <= b & ( A = ].b2,b.[ or A = ].b2,b.] or A = [.b2,b.] or A = [.b2,b.[ ) ) );

definition
let A be Interval;
assume A1: A <> {} ;
func A ^^ -> R_eal means :Def5: :: MEASURE6:def 5
ex a being R_eal st
( a <= it & ( A = ].a,it.[ or A = ].a,it.] or A = [.a,it.] or A = [.a,it.[ ) );
existence
ex b1, a being R_eal st
( a <= b1 & ( A = ].a,b1.[ or A = ].a,b1.] or A = [.a,b1.] or A = [.a,b1.[ ) )
proof end;
uniqueness
for b1, b2 being R_eal st ex a being R_eal st
( a <= b1 & ( A = ].a,b1.[ or A = ].a,b1.] or A = [.a,b1.] or A = [.a,b1.[ ) ) & ex a being R_eal st
( a <= b2 & ( A = ].a,b2.[ or A = ].a,b2.] or A = [.a,b2.] or A = [.a,b2.[ ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines ^^ MEASURE6:def 5 :
for A being Interval st A <> {} holds
for b2 being R_eal holds
( b2 = A ^^ iff ex a being R_eal st
( a <= b2 & ( A = ].a,b2.[ or A = ].a,b2.] or A = [.a,b2.] or A = [.a,b2.[ ) ) );

theorem Th48: :: MEASURE6:48
for A being Interval st A is open_interval & A <> {} holds
( ^^ A <= A ^^ & A = ].(^^ A),(A ^^ ).[ )
proof end;

theorem Th49: :: MEASURE6:49
for A being Interval st A is closed_interval & A <> {} holds
( ^^ A <= A ^^ & A = [.(^^ A),(A ^^ ).] )
proof end;

theorem Th50: :: MEASURE6:50
for A being Interval st A is right_open_interval & A <> {} holds
( ^^ A <= A ^^ & A = [.(^^ A),(A ^^ ).[ )
proof end;

theorem Th51: :: MEASURE6:51
for A being Interval st A is left_open_interval & A <> {} holds
( ^^ A <= A ^^ & A = ].(^^ A),(A ^^ ).] )
proof end;

theorem Th52: :: MEASURE6:52
for A being Interval st A <> {} holds
( ^^ A <= A ^^ & ( A = ].(^^ A),(A ^^ ).[ or A = ].(^^ A),(A ^^ ).] or A = [.(^^ A),(A ^^ ).] or A = [.(^^ A),(A ^^ ).[ ) )
proof end;

theorem :: MEASURE6:53
canceled;

theorem Th54: :: MEASURE6:54
for A being Interval
for a being real number st a in A holds
( ^^ A <= R_EAL a & R_EAL a <= A ^^ )
proof end;

theorem Th55: :: MEASURE6:55
for A, B being Interval
for a, b being real number st a in A & b in B & A ^^ <= ^^ B holds
a <= b
proof end;

theorem :: MEASURE6:56
for A being Interval
for a being R_eal st a in A holds
( ^^ A <= a & a <= A ^^ )
proof end;

theorem Th57: :: MEASURE6:57
for A being Interval st A <> {} holds
for a being R_eal st ^^ A < a & a < A ^^ holds
a in A
proof end;

theorem :: MEASURE6:58
for A, B being Interval st A ^^ = ^^ B & ( A ^^ in A or ^^ B in B ) holds
A \/ B is Interval
proof end;

definition
let A be Subset of REAL ;
let x be real number ;
func x + A -> Subset of REAL means :Def6: :: MEASURE6:def 6
for y being Real holds
( y in it iff ex z being Real st
( z in A & y = x + z ) );
existence
ex b1 being Subset of REAL st
for y being Real holds
( y in b1 iff ex z being Real st
( z in A & y = x + z ) )
proof end;
uniqueness
for b1, b2 being Subset of REAL st ( for y being Real holds
( y in b1 iff ex z being Real st
( z in A & y = x + z ) ) ) & ( for y being Real holds
( y in b2 iff ex z being Real st
( z in A & y = x + z ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines + MEASURE6:def 6 :
for A being Subset of REAL
for x being real number
for b3 being Subset of REAL holds
( b3 = x + A iff for y being Real holds
( y in b3 iff ex z being Real st
( z in A & y = x + z ) ) );

theorem Th59: :: MEASURE6:59
for A being Subset of REAL
for x being real number holds (- x) + (x + A) = A
proof end;

theorem :: MEASURE6:60
for x being real number
for A being Subset of REAL st A = REAL holds
x + A = A
proof end;

theorem :: MEASURE6:61
for x being real number holds x + {} = {}
proof end;

theorem Th62: :: MEASURE6:62
for A being Interval
for x being real number holds
( A is open_interval iff x + A is open_interval )
proof end;

theorem Th63: :: MEASURE6:63
for A being Interval
for x being real number holds
( A is closed_interval iff x + A is closed_interval )
proof end;

theorem Th64: :: MEASURE6:64
for A being Interval
for x being real number holds
( A is right_open_interval iff x + A is right_open_interval )
proof end;

theorem Th65: :: MEASURE6:65
for A being Interval
for x being real number holds
( A is left_open_interval iff x + A is left_open_interval )
proof end;

theorem Th66: :: MEASURE6:66
for A being Interval
for x being real number holds x + A is Interval
proof end;

registration
let A be Interval;
let x be real number ;
cluster x + A -> interval ;
correctness
coherence
x + A is interval
;
by Th66;
end;

theorem :: MEASURE6:67
for A being Interval
for x being real number holds vol A = vol (x + A)
proof end;