let X1, X2 be non empty set ; :: thesis: for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let M2 be sigma_Measure of S2; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for A being Element of S1
for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let A be Element of S1; :: thesis: for B being Element of S2
for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let B be Element of S2; :: thesis: for x being Element of X1
for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let x be Element of X1; :: thesis: for y being Element of X2 st E = [:A,B:] holds
( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
let y be Element of X2; :: thesis: ( E = [:A,B:] implies ( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) ) )
assume A1: E = [:A,B:] ; :: thesis: ( Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) & Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) )
then A2: Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . B) * ((chi (A,X1)) . x) by Th47;
A3: (M2 . B) * ((chi (A,X1)) . x) = M2 . (Measurable-X-section (E,x)) by A1, Th48;
A4: Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . A) * ((chi (B,X2)) . y) by A1, Th49;
A5: (M1 . A) * ((chi (B,X2)) . y) = M1 . (Measurable-Y-section (E,y)) by A1, Th50;
thus Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) :: thesis: Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y)

proof

per cases ( x in A or not x in A ) ;

suppose x in A ; :: thesis: Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x)

then (chi (A,X1)) . x = 1 by FUNCT_3:def 3;
hence Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) by A2, A3, XXREAL_3:81; :: thesis: verum

end;

suppose not x in A ; :: thesis: Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x)

then (chi (A,X1)) . x = 0 by FUNCT_3:def 3;
hence Integral (M2,(ProjMap1 ((chi (E,[:X1,X2:])),x))) = (M2 . (Measurable-X-section (E,x))) * ((chi (A,X1)) . x) by A2; :: thesis: verum

end;

end;

end;

per cases ( y in B or not y in B ) ;

suppose y in B ; :: thesis: Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y)

then (chi (B,X2)) . y = 1 by FUNCT_3:def 3;
hence Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) by A4, A5, XXREAL_3:81; :: thesis: verum

end;

suppose not y in B ; :: thesis: Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y)

then (chi (B,X2)) . y = 0 by FUNCT_3:def 3;
hence Integral (M1,(ProjMap2 ((chi (E,[:X1,X2:])),y))) = (M1 . (Measurable-Y-section (E,y))) * ((chi (B,X2)) . y) by A4; :: thesis: verum

end;

end;