let X1, X2 be non empty set ; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL
for x being Element of X1
for y being Element of X2 holds
( ProjPMap1 (|.f.|,x) = |.(ProjPMap1 (f,x)).| & ProjPMap2 (|.f.|,y) = |.(ProjPMap2 (f,y)).| )
let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: for x being Element of X1
for y being Element of X2 holds
( ProjPMap1 (|.f.|,x) = |.(ProjPMap1 (f,x)).| & ProjPMap2 (|.f.|,y) = |.(ProjPMap2 (f,y)).| )
let x be Element of X1; :: thesis: for y being Element of X2 holds
( ProjPMap1 (|.f.|,x) = |.(ProjPMap1 (f,x)).| & ProjPMap2 (|.f.|,y) = |.(ProjPMap2 (f,y)).| )
let y be Element of X2; :: thesis: ( ProjPMap1 (|.f.|,x) = |.(ProjPMap1 (f,x)).| & ProjPMap2 (|.f.|,y) = |.(ProjPMap2 (f,y)).| )
A1: |.f.| = (max+ f) + (max- f) by MESFUNC2:24;
then ProjPMap1 (|.f.|,x) = (ProjPMap1 ((max+ f),x)) + (ProjPMap1 ((max- f),x)) by MESFUN12:44
.= (max+ (ProjPMap1 (f,x))) + (ProjPMap1 ((max- f),x)) by MESFUN12:45
.= (max+ (ProjPMap1 (f,x))) + (max- (ProjPMap1 (f,x))) by MESFUN12:45 ;
hence ProjPMap1 (|.f.|,x) = |.(ProjPMap1 (f,x)).| by MESFUNC2:24; :: thesis: ProjPMap2 (|.f.|,y) = |.(ProjPMap2 (f,y)).|
ProjPMap2 (|.f.|,y) = (ProjPMap2 ((max+ f),y)) + (ProjPMap2 ((max- f),y)) by A1, MESFUN12:44
.= (max+ (ProjPMap2 (f,y))) + (ProjPMap2 ((max- f),y)) by MESFUN12:46
.= (max+ (ProjPMap2 (f,y))) + (max- (ProjPMap2 (f,y))) by MESFUN12:46 ;
hence ProjPMap2 (|.f.|,y) = |.(ProjPMap2 (f,y)).| by MESFUNC2:24; :: thesis: verum