
theorem
  for T being TopStruct for S being Subset of T holds S is trivial
  implies S is strong closed_under_lines
proof
  let T be TopStruct;
  let S be Subset of T;
  assume
A1: S is trivial;
  thus S is strong
  proof
    let x,y be Point of T;
    assume
A2: x in S & y in S;
    thus x,y are_collinear
    proof
      per cases;
      suppose
        x=y;
        hence thesis;
      end;
      suppose
        x<>y;
        then 2 c= card S by A2,Th2;
        hence thesis by A1,Th4;
      end;
    end;
  end;
  thus S is closed_under_lines
  proof
    let l be Block of T;
A3: card (l /\ S) c= card S by CARD_1:11,XBOOLE_1:17;
    assume 2 c= card (l /\ S);
    then 2 c= card S by A3;
    hence thesis by A1,Th4;
  end;
end;
