
theorem Th41:
for X1,X2 be non empty set, A,Z be set, f be PartFunc of [:X1,X2:],Z,
 x be Element of X1 holds X-section(f"A,x) = ProjPMap1(f,x)"A
proof
    let X,Y be non empty set, A,Z be set, f be PartFunc of [:X,Y:],Z,
    x be Element of X;
    reconsider E = f"A as Subset of [:X,Y:];
    now let y be object;
     assume y in X-section(f"A,x); then
     y in {y where y is Element of Y: [x,y] in E} by MEASUR11:def 4; then
     consider y1 be Element of Y such that
A1:   y1 = y & [x,y1] in E;
A2:  [x,y] in dom f & f.[x,y] in A by A1,FUNCT_1:def 7; then
     y in {y where y is Element of Y: [x,y] in dom f} by A1; then
     y in X-section(dom f,x) by MEASUR11:def 4; then
A3:  y in dom(ProjPMap1(f,x)) by Def3;

     ProjPMap1(f,x).y1 = f.(x,y1) by A1,A2,Def3;
     hence y in ProjPMap1(f,x)"A by A1,A2,A3,FUNCT_1:def 7;
    end; then
A4: X-section(f"A,x) c= ProjPMap1(f,x)"A;

    now let y be object;
     assume y in ProjPMap1(f,x)"A; then
A5:  y in dom(ProjPMap1(f,x)) & ProjPMap1(f,x).y in A by FUNCT_1:def 7; then
     y in X-section(dom f,x) by Def3; then
     y in {y where y is Element of Y: [x,y] in dom f} by MEASUR11:def 4; then
     consider y1 be Element of Y such that
A6:   y1 = y & [x,y1] in dom f;

     f.(x,y1) in A by A5,A6,Def3; then
     [x,y1] in f"A by A6,FUNCT_1:def 7; then
     y in {y where y is Element of Y: [x,y] in f"A} by A6;
     hence y in X-section(f"A,x) by MEASUR11:def 4;
    end; then
    ProjPMap1(f,x)"A c= X-section(f"A,x);
    hence X-section(f"A,x) = ProjPMap1(f,x)"A by A4;
end;
