
theorem Th32:
for X,Y be non empty set, f be PartFunc of [:X,Y:],REAL,
 x be Element of X, y be Element of Y holds
   ProjPMap1(|.f.|,x) = |. ProjPMap1(f,x) .|
 & ProjPMap2(|.f.|,y) = |. ProjPMap2(f,y) .|
proof
    let X,Y be non empty set, f be PartFunc of [:X,Y:],REAL,
    x be Element of X, y be Element of Y;

A1: dom f = dom |.f.| by VALUED_1:def 11;
A2: dom ProjPMap1(f,x) = dom |. ProjPMap1(f,x) .| &
    dom ProjPMap2(f,y) = dom |. ProjPMap2(f,y) .| by VALUED_1:def 11;

A3: dom ProjPMap1(|.f.|,x) = X-section(dom f,x) by A1,MESFUN12:def 3; then
A4: dom ProjPMap1(|.f.|,x) = dom |. ProjPMap1(f,x) .| by A2,MESFUN12:def 3;

    for y be Element of Y st y in dom ProjPMap1(|.f.|,x) holds
     ProjPMap1(|.f.|,x).y = (|. ProjPMap1(f,x) .|).y
    proof
     let y be Element of Y;
     assume
A5:   y in dom ProjPMap1(|.f.|,x); then
     y in {y where y is Element of Y: [x,y] in dom f}
       by A3,MEASUR11:def 4; then
A6:  ex y1 be Element of Y st y1 = y & [x,y1] in dom f; then
A7:  ProjPMap1(|.f.|,x).y = (|.f.|).(x,y) by A1,MESFUN12:def 3;

     ProjPMap1(f,x).y = f.(x,y) by A6,MESFUN12:def 3; then
     (|. ProjPMap1(f,x) .|).y = |. f.([x,y]) .| by A5,A4,VALUED_1:def 11;
     hence thesis by A1,A7,A6,VALUED_1:def 11;
    end;
    hence ProjPMap1(|.f.|,x) = |. ProjPMap1(f,x) .| by A4,PARTFUN1:5;

A8: dom ProjPMap2(|.f.|,y) = Y-section(dom f,y) by A1,MESFUN12:def 4; then
A9: dom ProjPMap2(|.f.|,y) = dom |. ProjPMap2(f,y) .| by A2,MESFUN12:def 4;

    for x be Element of X st x in dom ProjPMap2(|.f.|,y) holds
     ProjPMap2(|.f.|,y).x = (|. ProjPMap2(f,y) .|).x
    proof
     let x be Element of X;
     assume
A10:   x in dom ProjPMap2(|.f.|,y); then
     x in {x where x is Element of X: [x,y] in dom f}
       by A8,MEASUR11:def 5; then
A11:  ex x1 be Element of X st x1 = x & [x1,y] in dom f; then
A12:  ProjPMap2(|.f.|,y).x = (|.f.|).(x,y) by A1,MESFUN12:def 4;

     ProjPMap2(f,y).x = f.(x,y) by A11,MESFUN12:def 4; then
     (|. ProjPMap2(f,y) .|).x = |. f.([x,y]) .| by A10,A9,VALUED_1:def 11;
     hence thesis by A1,A12,A11,VALUED_1:def 11;
    end;
    hence ProjPMap2(|.f.|,y) = |. ProjPMap2(f,y) .| by A9,PARTFUN1:5;
end;
