
theorem
  for S being non empty non void TopStruct for f being Collineation of S
  for X,Y being Subset of S st X is non trivial & Y is non trivial & X,Y
  are_joinable holds f"X,f"Y are_joinable
proof
  let S be non empty non void TopStruct;
  let f be Collineation of S;
  let X,Y be Subset of S;
  reconsider g=f" as Collineation of S by Th13;
  assume that
A1: X is non trivial and
A2: Y is non trivial and
A3: X,Y are_joinable;
A4: f is bijective by Def4;
  then
A5: rng f = [#]S by FUNCT_2:def 3;
  then
A6: f"Y = g.:Y by A4,TOPS_2:55;
  f"X = g.:X by A4,A5,TOPS_2:55;
  hence thesis by A6,A1,A2,A3,Th20;
end;
