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
  id X c= id Y iff X c= Y
proof
  thus id X c= id Y implies X c= Y
  proof
    assume
A1: id X c= id Y;
    let x be object;
    assume x in X;
    then [x,x] in id X by RELAT_1:def 10;
    hence thesis by A1,RELAT_1:def 10;
  end;
  assume
A2: X c= Y;
  let z be object;
  assume
A3: z in id X;
  then consider x,x9 being object such that
A4: x in X and
  x9 in X and
A5: z = [x,x9] by ZFMISC_1:84;
  x = x9 by A3,A5,RELAT_1:def 10;
  hence thesis by A2,A4,A5,RELAT_1:def 10;
end;
