reserve X for set, a,b,c,x,y,z for object;
reserve P,R for Relation;

theorem
  R is reflexive implies dom R = dom(R~) & rng R = rng(R~)
proof
  assume
A1: R is reflexive; then
A2: R is_reflexive_in field R;
A3: R~ is_reflexive_in field(R~) by A1,Def9;
  now
    let x be object;
A4: now
      assume x in dom(R~);
      then x in field(R~) by XBOOLE_0:def 3;
      then [x,x] in R~ by A1,Def1,Def9;
      then [x,x] in R by RELAT_1:def 7;
      hence x in dom R by XTUPLE_0:def 12;
    end;
    now
      assume x in dom R;
      then x in field R by XBOOLE_0:def 3;
      then [x,x] in R by A1,Def1;
      then [x,x] in R~ by RELAT_1:def 7;
      hence x in dom(R~) by XTUPLE_0:def 12;
    end;
    hence x in dom R iff x in dom(R~) by A4;
  end;
  hence dom R = dom(R~) by TARSKI:2;
  now
    let x be object;
A5: now
      assume x in rng(R~);
      then x in field(R~) by XBOOLE_0:def 3;
      then [x,x] in R~ by A3;
      then [x,x] in R by RELAT_1:def 7;
      hence x in rng R by XTUPLE_0:def 13;
    end;
    now
      assume x in rng R;
      then x in field R by XBOOLE_0:def 3;
      then [x,x] in R by A2;
      then [x,x] in R~ by RELAT_1:def 7;
      hence x in rng(R~) by XTUPLE_0:def 13;
    end;
    hence x in rng R iff x in rng(R~) by A5;
  end;
  hence thesis by TARSKI:2;
end;
