
theorem Th31:
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(R_EAL f,x) = R_EAL(ProjPMap1(f,x))
 & ProjPMap1(|.R_EAL f.|,x) = |. R_EAL(ProjPMap1(f,x)) .|
 & ProjPMap2(R_EAL f,y) = R_EAL(ProjPMap2(f,y))
 & ProjPMap2(|.R_EAL f.|,y) = |. R_EAL(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(R_EAL f) by MESFUNC5:def 7; then
A2: dom ProjPMap1(R_EAL f,x) = X-section(dom f,x)
  & dom ProjPMap2(R_EAL f,y) = Y-section(dom f,y) by MESFUN12:def 3,def 4;
    dom R_EAL(ProjPMap1(f,x)) = dom ProjPMap1(f,x)
  & dom R_EAL(ProjPMap2(f,y)) = dom ProjPMap2(f,y) by MESFUNC5:def 7; then
A3: dom R_EAL(ProjPMap1(f,x)) = X-section(dom f,x)
  & dom R_EAL(ProjPMap2(f,y)) = Y-section(dom f,y) by MESFUN12:def 3,def 4;

    for y be Element of Y st y in dom ProjPMap1(R_EAL f,x) holds
     (ProjPMap1(R_EAL f,x)).y = (R_EAL(ProjPMap1(f,x))).y
    proof
     let y be Element of Y;
     assume y in dom ProjPMap1(R_EAL f,x); then
A4:  y in X-section(dom f,x) by A1,MESFUN12:def 3;
     X-section(dom f,x) = {y where y is Element of Y: [x,y] in dom f}
       by MEASUR11:def 4; then
A5:  ex y0 be Element of Y st y0 = y & [x,y0] in dom f by A4; then
     (ProjPMap1(R_EAL f,x)).y = (R_EAL f).(x,y) by A1,MESFUN12:def 3; then
A6:  (ProjPMap1(R_EAL f,x)).y = f.(x,y) by MESFUNC5:def 7;

     (R_EAL(ProjPMap1(f,x))).y = (ProjPMap1(f,x)).y by MESFUNC5:def 7;
     hence thesis by A6,A5,MESFUN12:def 3;
    end;
    hence
    ProjPMap1(R_EAL f,x) = R_EAL(ProjPMap1(f,x)) by A2,A3,PARTFUN1:5;
    hence ProjPMap1(|.R_EAL f.|,x) = |. R_EAL(ProjPMap1(f,x)) .|
      by MESFUN13:7;

    for x be Element of X st x in dom ProjPMap2(R_EAL f,y) holds
     (ProjPMap2(R_EAL f,y)).x = (R_EAL(ProjPMap2(f,y))).x
    proof
     let x be Element of X;
     assume x in dom ProjPMap2(R_EAL f,y); then
A7:  x in Y-section(dom f,y) by A1,MESFUN12:def 4;
     Y-section(dom f,y) = {x where x is Element of X: [x,y] in dom f}
       by MEASUR11:def 5; then
A8:  ex x0 be Element of X st x0 = x & [x0,y] in dom f by A7; then
     (ProjPMap2(R_EAL f,y)).x = (R_EAL f).(x,y) by A1,MESFUN12:def 4; then
A9:  (ProjPMap2(R_EAL f,y)).x = f.(x,y) by MESFUNC5:def 7;

     (R_EAL(ProjPMap2(f,y))).x = (ProjPMap2(f,y)).x by MESFUNC5:def 7;
     hence thesis by A9,A8,MESFUN12:def 4;
    end;
    hence
    ProjPMap2(R_EAL f,y) = R_EAL(ProjPMap2(f,y)) by A2,A3,PARTFUN1:5;
    hence ProjPMap2(|.R_EAL f.|,y) = |. R_EAL(ProjPMap2(f,y)) .|
      by MESFUN13:7;
end;
