
theorem Th10:
  for S being TopStruct for X being Subset of S st X is
  closed_under_lines strong holds X,X are_joinable
proof
  let S be TopStruct;
  let X be Subset of S;
  assume
A1: X is closed_under_lines strong;
  reconsider f = <*X*> as FinSequence of bool the carrier of S;
  take f;
  thus X = f.1;
  len f = 1 by FINSEQ_1:40;
  hence X = f.(len f);
  thus for W being Subset of S st W in rng f holds W is closed_under_lines
  strong
  proof
    let W be Subset of S;
    assume W in rng f;
    then W in {X} by FINSEQ_1:38;
    hence thesis by A1,TARSKI:def 1;
  end;
  let i be Element of NAT;
  assume that
A2: 1 <= i and
A3: i < len f;
  thus thesis by A2,A3,FINSEQ_1:40;
end;
