theorem Th47:
  dom f c= [:X,Y:] implies rng ~f = rng f
proof
  assume
A1: dom f c= [:X,Y:];
  thus rng ~f c= rng f by Th41;
  let y be object;
  assume y in rng f;
  then consider x being object such that
A2: x in dom f and
A3: y = f.x by FUNCT_1:def 3;
  consider x1,y1 being object such that
  x1 in X and
  y1 in Y and
A4: x =[x1,y1] by A1,A2,ZFMISC_1:84;
A5: [y1,x1] in dom ~f by A2,A4,Th42;
  y = f.(x1,y1) by A3,A4
    .= (~f).(y1,x1) by A2,A4,Def2;
  hence thesis by A5,FUNCT_1:def 3;
end;
