reserve X for set;
reserve X,X1,X2 for non empty set;

theorem Th27:
for f being PartFunc of [:X1,X2:],ExtREAL,
    x being Element of X1, y being Element of X2
holds ProjPMap1(|.f.|,x) = |.ProjPMap1(f,x).|
    & ProjPMap2(|.f.|,y) = |.ProjPMap2(f,y).|
proof
    let f be PartFunc of [:X1,X2:],ExtREAL,
    x be Element of X1, y be Element of X2;
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;
    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;
end;
