
theorem
  for S being non empty non void TopStruct for f being Collineation of S
  for X being Subset of S st X is closed_under_lines holds f"X is
  closed_under_lines
proof
  let S be non empty non void TopStruct;
  let f be Collineation of S;
  reconsider g=f" as Collineation of S by Th13;
  let X be Subset of S;
  assume X is closed_under_lines;
  then
A1: g.:X is closed_under_lines by Th18;
A2: f is bijective by Def4;
  then rng f = [#]S by FUNCT_2:def 3;
  hence thesis by A2,A1,TOPS_2:55;
end;
