reserve r,s,t,u for Real;

theorem Th41:
  for X being LinearTopSpace, V1,V2 being bounded Subset of X
  holds V1 \/ V2 is bounded
proof
  let X be non empty LinearTopSpace, V1,V2 be bounded Subset of X;
  thus thesis
  proof
    let V be a_neighborhood of 0.X;
    consider s such that
A1: s > 0 and
A2: for t st t > s holds V1 c= t*V by Def12;
    consider r such that
A3: r > 0 and
A4: for t st t > r holds V2 c= t*V by Def12;
    per cases;
    suppose
A5:   r <= s;
      take s;
      thus s > 0 by A1;
      let t such that
A6:   t > s;
      t > r by A5,A6,XXREAL_0:2;
      then
A7:   V2 c= t*V by A4;
      V1 c= t*V by A2,A6;
      hence thesis by A7,XBOOLE_1:8;
    end;
    suppose
A8:   r > s;
      take r;
      thus r > 0 by A3;
      let t such that
A9:   t > r;
      t > s by A8,A9,XXREAL_0:2;
      then
A10:  V1 c= t*V by A2;
      V2 c= t*V by A4,A9;
      hence thesis by A10,XBOOLE_1:8;
    end;
  end;
end;
