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 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;
