theorem Th52:
  dom f c= [:X,Y:] implies ~~f = f
proof
  assume
A1: dom f c= [:X,Y:];
A2: ~~f c= f by Th51;
  dom ~~ f = dom f
  proof
    thus dom ~~f c= dom f by A2,RELAT_1:11;
    let z be object;
    assume
A3: z in dom f;
    then consider x,y being object such that
    x in X and
    y in Y and
A4: z = [x,y] by A1,ZFMISC_1:84;
    [y,x] in dom ~f by A3,A4,Th42;
    hence thesis by A4,Th42;
  end;
  hence thesis by Th51,GRFUNC_1:3;
end;
