
theorem Th3:
  for A being set, f being Function holds f.:id A = (~f).:id A
proof
  let A be set, f be Function;
  thus f.:id A c= (~f).:id A
  proof
    let y be object;
    assume y in f.:id A;
    then consider x being object such that
A1: x in dom f and
A2: x in id A and
A3: y = f.x by FUNCT_1:def 6;
    consider x1,x2 being object such that
A4: x = [x1,x2] by A2,RELAT_1:def 1;
A5: x1 = x2 by A2,A4,RELAT_1:def 10;
    then
A6: x in dom~f by A1,A4,FUNCT_4:42;
    then f.(x1,x2) = (~f).(x1,x2) by A4,A5,FUNCT_4:43;
    hence thesis by A2,A3,A4,A6,FUNCT_1:def 6;
  end;
  let y be object;
  assume y in (~f).:id A;
  then consider x being object such that
A7: x in dom~f and
A8: x in id A and
A9: y = (~f).x by FUNCT_1:def 6;
  consider x1,x2 being object such that
A10: x = [x1,x2] by A8,RELAT_1:def 1;
A11: x1 = x2 by A8,A10,RELAT_1:def 10;
  then
A12: x in dom f by A7,A10,FUNCT_4:42;
  then ~f.(x1,x2) = f.(x1,x2) by A10,A11,FUNCT_4:def 2;
  hence thesis by A8,A9,A10,A12,FUNCT_1:def 6;
end;
