reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;

theorem
  for f being Function st f is one-to-one holds <:f,X,Y:>" = <:f",Y,X:>
proof
  let f be Function;
  assume
A1: f is one-to-one;
  then
A2: <:f,X,Y:> is one-to-one by Th37;
  for y being object holds y in dom (<:f,X,Y:>") iff y in dom <:f",Y,X:>
  proof let y be object;
    thus y in dom (<:f,X,Y:>") implies y in dom <:f",Y,X:>
    proof
      assume y in dom (<:f,X,Y:>");
      then
A3:   y in rng <:f,X,Y:> by A2,FUNCT_1:33;
      then consider x being object such that
A4:   x in dom <:f,X,Y:> and
A5:   y = <:f,X,Y:>.x by FUNCT_1:def 3;
A6:   f.x = y by A4,A5,Th26;
      then
A7:   y in Y by A4,Th24;
      rng <:f,X,Y:> c= rng f by Th23;
      then y in rng f by A3;
      then
A8:   y in dom(f") by A1,FUNCT_1:32;
      dom <:f,X,Y:> c= dom f by Th23;
      then (f").y = x by A1,A4,A6,FUNCT_1:32;
      hence thesis by A4,A8,A7,Th24;
    end;
    assume
A9: y in dom <:f",Y,X:>;
    dom <:f",Y,X:> c= dom (f") by Th23;
    then y in dom(f") by A9;
    then y in rng f by A1,FUNCT_1:33;
    then consider x being object such that
A10: x in dom f and
A11: y = f.x by FUNCT_1:def 3;
    x =(f").(f.x) by A1,A10,FUNCT_1:34;
    then x in X by A9,A11,Th24;
    then x in dom <:f,X,Y:> by A9,A10,A11,Th24;
    then <:f,X,Y:>.x in rng <:f,X,Y:> & <:f,X,Y:>.x = f.x by Th26,FUNCT_1:def 3
;
    hence thesis by A2,A11,FUNCT_1:33;
  end;
  then
A12: dom (<:f,X,Y:>") = dom <:f",Y,X:> by TARSKI:2;
  for y being object st y in dom <:f",Y,X:>
holds <:f",Y,X:>.y = (<:f,X,Y:>").y
  proof
    let y be object;
A13: rng <:f,X,Y:> c= rng f by Th23;
    assume
A14: y in dom <:f",Y,X:>;
    then y in rng <:f,X,Y:> by A2,A12,FUNCT_1:33;
    then consider x being object such that
A15: x in dom f and
A16: y = f.x by A13,FUNCT_1:def 3;
A17: x =(f").(f.x) by A1,A15,FUNCT_1:34;
    then x in X by A14,A16,Th24;
    then x in dom<:f,X,Y:> by A14,A15,A16,Th24;
    then (<:f,X,Y:>").(<:f,X,Y:>.x) = x & <:f,X,Y:>.x = f.x by A2,Th26,
FUNCT_1:34;
    hence thesis by A14,A16,A17,Th26;
  end;
  hence thesis by A12;
end;
