theorem Th42:
 for x,y being object holds [x,y] in dom f iff [y,x] in dom ~f
proof let x,y be object;
  thus [x,y] in dom f implies [y,x] in dom ~f by Def2;
  assume [y,x] in dom ~f;
  then consider x1,y1 being object such that
A1: [y,x] = [y1,x1] and
A2: [x1,y1] in dom f by Def2;
  x1 = x by A1,XTUPLE_0:1;
  hence thesis by A1,A2,XTUPLE_0:1;
end;
