reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

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;
