theorem Th41:
  rng ~f c= rng f
proof
  let y be object;
  assume y in rng ~f;
  then consider x being object such that
A1: x in dom ~f and
A2: y = (~f).x by FUNCT_1:def 3;
  consider x1,x2 being object such that
A3: x = [x2,x1] and
A4: [x1,x2] in dom f by A1,Def2;
  y = (~f).(x2,x1) by A2,A3
    .= f.(x1,x2) by A4,Def2;
  hence thesis by A4,FUNCT_1:def 3;
end;
