
theorem Th14:
  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;
A4: dom f = the carrier of S by FUNCT_2:def 1;
  then
A5: f.x in f.:X by A1,FUNCT_1:def 6;
A6: f.y in f.:X by A4,A2,FUNCT_1:def 6;
  f is bijective by Def4;
  then f.x <> f.y by A4,A1,A2,A3,FUNCT_1:def 4;
  then 2 c= card(f.:X) by A5,A6,PENCIL_1:2;
  hence thesis by PENCIL_1:4;
end;
