reserve x,y for set;

theorem Th44:
  for f, g being Function st f c= g holds ~f c= ~g
proof
  let f, g be Function such that
A1: f c= g;
  let x,y be object;
  assume
A2: [x,y] in ~f;
  then x in dom ~f by XTUPLE_0:def 12;
  then consider z2,z1 being object such that
A3: x = [z1,z2] and
A4: [z2,z1] in dom f by FUNCT_4:def 2;
  y = (~f).(z1,z2) by A2,A3,FUNCT_1:1
    .= f.(z2,z1) by A4,FUNCT_4:def 2;
  then
A5: [[z2,z1],y] in f by A4,FUNCT_1:1;
  then
A6: [z2,z1] in dom g by A1,FUNCT_1:1;
  y = g.(z2,z1) by A1,A5,FUNCT_1:1;
  then
A7: y = (~g).(z1,z2) by A6,FUNCT_4:def 2;
  x in dom ~g by A3,A6,FUNCT_4:def 2;
  hence thesis by A3,A7,FUNCT_1:1;
end;
