
theorem
  for S being non empty TopStruct for f being Collineation of S for X
  being Subset of S st X is non trivial holds f"X is non trivial
proof
  let S be non empty TopStruct;
  let f be Collineation of S;
  let X be Subset of S;
  assume X is non trivial;
  then 2 c= card X by PENCIL_1:4;
  then consider x,y being object such that
A1: x in X and
A2: y in X and
A3: x<>y by PENCIL_1:2;
  f is bijective by Def4;
  then
A4: rng f = the carrier of S by FUNCT_2:def 3;
  then consider fx being object such that
A5: fx in dom f and
A6: f.fx = x by A1,FUNCT_1:def 3;
  consider fy being object such that
A7: fy in dom f and
A8: f.fy = y by A4,A2,FUNCT_1:def 3;
A9: fy in f"X by A2,A7,A8,FUNCT_1:def 7;
  fx in f"X by A1,A5,A6,FUNCT_1:def 7;
  then 2 c= card(f"X) by A3,A6,A8,A9,PENCIL_1:2;
  hence thesis by PENCIL_1:4;
end;
