theorem Th49:
  x in Union F1 iff ex k st k in dom F1 & x in F1.k
proof
  set Y = union rng F1;
A1: for x holds x in Y iff ex k st k in dom F1 & x in F1.k
  proof
    let x;
    thus x in Y implies ex k st k in dom F1 & x in F1.k
    proof
      assume x in Y;
      then consider Z such that
A2:   x in Z and
A3:   Z in rng F1 by TARSKI:def 4;
      ex i being Nat st i in dom F1 & Z = F1.i by A3,FINSEQ_2:10;
      hence thesis by A2;
    end;
    thus (ex k st k in dom F1 & x in F1.k) implies x in Y
    proof
      given i such that
A4:   i in dom F1 and
A5:   x in F1.i;
      F1.i in rng F1 by A4,FUNCT_1:def 3;
      hence thesis by A5,TARSKI:def 4;
    end;
  end;
  Y = Union F1 by CARD_3:def 4;
  hence thesis by A1;
end;
