reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th14:
  rng R = dom(R~) & dom R = rng(R~)
proof
  thus rng R c= dom(R~)
  proof
    let u be object;
    assume u in rng R;
    then consider x being object such that
A1: [x,u] in R by XTUPLE_0:def 13;
    [u,x] in R~ by A1,Def5;
    hence thesis by XTUPLE_0:def 12;
  end;
  thus dom(R~) c= rng R
  proof
    let u be object;
    assume u in dom(R~);
    then consider x being object such that
A2: [u,x] in R~ by XTUPLE_0:def 12;
    [x,u] in R by A2,Def5;
    hence thesis by XTUPLE_0:def 13;
  end;
  thus dom R c= rng(R~)
  proof
    let u be object;
    assume u in dom R;
    then consider x being object such that
A3: [u,x] in R by XTUPLE_0:def 12;
    [x,u] in R~ by A3,Def5;
    hence thesis by XTUPLE_0:def 13;
  end;
  let u be object;
  assume u in rng(R~);
  then consider x being object such that
A4: [x,u] in R~ by XTUPLE_0:def 13;
  [u,x] in R by A4,Def5;
  hence thesis by XTUPLE_0:def 12;
end;
