let X1, X2 be non empty set ; :: thesis: for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
Y-vol (E,M2) = Integral2 (M2,(chi (E,[:X1,X2:])))
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
Y-vol (E,M2) = Integral2 (M2,(chi (E,[:X1,X2:])))
let S2 be SigmaField of X2; :: thesis: for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
Y-vol (E,M2) = Integral2 (M2,(chi (E,[:X1,X2:])))
let M2 be sigma_Measure of S2; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
Y-vol (E,M2) = Integral2 (M2,(chi (E,[:X1,X2:])))
let A be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: ( M2 is sigma_finite implies Y-vol (A,M2) = Integral2 (M2,(chi (A,[:X1,X2:]))) )
assume a1: M2 is sigma_finite ; :: thesis: Y-vol (A,M2) = Integral2 (M2,(chi (A,[:X1,X2:])))

now :: thesis: for x being Element of X1 holds (Y-vol (A,M2)) . x = (Integral2 (M2,(chi (A,[:X1,X2:])))) . x

let x be Element of X1; :: thesis: (Y-vol (A,M2)) . x = (Integral2 (M2,(chi (A,[:X1,X2:])))) . x
A1: (Y-vol (A,M2)) . x = Integral (M2,(chi ((Measurable-X-section (A,x)),X2))) by a1, Th62;
ProjPMap1 ((chi (A,[:X1,X2:])),x) = chi ((Measurable-X-section (A,x)),X2) by Th63;
hence (Y-vol (A,M2)) . x = (Integral2 (M2,(chi (A,[:X1,X2:])))) . x by A1, Def8; :: thesis: verum

end;

hence Y-vol (A,M2) = Integral2 (M2,(chi (A,[:X1,X2:]))) by FUNCT_2:def 8; :: thesis: verum