reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace, F being Subset-Family of X st F is finite
  & F = the set of all P where P is bounded Subset of X holds union F is
  bounded
proof
  let X be LinearTopSpace, F be Subset-Family of X such that
A1: F is finite and
A2: F = the set of all P where P is bounded Subset of X;
  defpred P[set] means ex A being Subset of X st A = union $1 & A is bounded;
A3: now
    let P be Subset of X;
    assume P in F;
    then ex A being bounded Subset of X st P=A by A2;
    hence P is bounded;
  end;
A4: for x,B being set st x in F & B c= F & P[B] holds P[B \/ {x}]
  proof
    let x,B be set such that
A5: x in F and
    B c= F and
A6: P[B];
    consider A being Subset of X such that
A7: A = union B & A is bounded by A6;
    reconsider x as Subset of X by A5;
A8: union (B \/ {x}) = union B \/ union {x} by ZFMISC_1:78
      .= union B \/ x by ZFMISC_1:25;
A9: x is bounded by A3,A5;
    ex Y being Subset of X st Y = union (B \/ {x}) & Y is bounded
    proof
      take A \/ x;
      thus thesis by A7,A8,A9,Th41;
    end;
    hence thesis;
  end;
  {}X = union {} by ZFMISC_1:2;
  then
A10: P[{}];
  P[F] from FINSET_1:sch 2(A1,A10,A4);
  hence thesis;
end;
