reserve X for RealUnitarySpace;
reserve x for Point of X;
reserve i, n for Nat;

theorem
  for Y be finite Subset of X st Y is non empty holds Y is summable_set
proof
  let Y be finite Subset of X such that
A1: Y is non empty;
  set x = setsum Y;
  now
    let e be Real such that
A2: e >0;
    ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be
    finite Subset of X st Y0 c= Y1 & Y1 c= Y holds ||.x-setsum(Y1).|| < e
    proof
      take Y;
      now
        let Y1 be finite Subset of X;
        assume Y c= Y1 & Y1 c= Y;
        then Y1 = Y by XBOOLE_0:def 10;
        then x - setsum(Y1) = 0.X by RLVECT_1:15;
        hence ||.x-setsum(Y1).|| < e by A2,BHSP_1:26;
      end;
      hence thesis by A1;
    end;
    hence ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be
    finite Subset of X st Y0 c= Y1 & Y1 c= Y holds ||.x-setsum(Y1).|| < e;
  end;
  hence thesis;
end;
