
theorem Th42:
for X1,X2 be non empty set, A,Z be set, f be PartFunc of [:X1,X2:],Z,
 y be Element of X2 holds Y-section(f"A,y) = ProjPMap2(f,y)"A
proof
    let X,Y be non empty set, A,Z be set, f be PartFunc of [:X,Y:],Z,
    y be Element of Y;
    reconsider E = f"A as Subset of [:X,Y:];
    now let x be object;
     assume x in Y-section(f"A,y); then
     x in {x where x is Element of X: [x,y] in E} by MEASUR11:def 5; then
     consider x1 be Element of X such that
A1:   x1 = x & [x1,y] in E;
A2:  [x,y] in dom f & f.[x,y] in A by A1,FUNCT_1:def 7; then
     x in {x where x is Element of X: [x,y] in dom f} by A1; then
     x in Y-section(dom f,y) by MEASUR11:def 5; then
A3:  x in dom(ProjPMap2(f,y)) by Def4;

     ProjPMap2(f,y).x1 = f.(x1,y) by A1,A2,Def4;
     hence x in ProjPMap2(f,y)"A by A1,A2,A3,FUNCT_1:def 7;
    end; then
A4: Y-section(f"A,y) c= ProjPMap2(f,y)"A;

    now let x be object;
     assume x in ProjPMap2(f,y)"A; then
A5:  x in dom(ProjPMap2(f,y)) & ProjPMap2(f,y).x in A by FUNCT_1:def 7; then
     x in Y-section(dom f,y) by Def4; then
     x in {x where x is Element of X: [x,y] in dom f} by MEASUR11:def 5; then
     consider x1 be Element of X such that
A6:   x1 = x & [x1,y] in dom f;

     f.(x1,y) in A by A5,A6,Def4; then

     [x1,y] in f"A by A6,FUNCT_1:def 7; then
     x in {x where x is Element of X: [x,y] in f"A} by A6;
     hence x in Y-section(f"A,y) by MEASUR11:def 5;
    end; then
    ProjPMap2(f,y)"A c= Y-section(f"A,y);
    hence Y-section(f"A,y) = ProjPMap2(f,y)"A by A4;
end;
