
theorem Th1:
  for S being non empty non void TopStruct for f being Collineation
of S for p,q being Point of S st p,q are_collinear holds f.p,f.q are_collinear
proof
  let S be non empty non void TopStruct;
  let f be Collineation of S;
A1: dom f = the carrier of S by FUNCT_2:def 1;
  let p,q be Point of S;
  assume
A2: p,q are_collinear;
  per cases;
  suppose
    p=q;
    hence thesis by PENCIL_1:def 1;
  end;
  suppose
    p<>q;
    then consider B being Block of S such that
A3: {p,q} c= B by A2,PENCIL_1:def 1;
    q in B by A3,ZFMISC_1:32;
    then
A4: f.q in f.:B by A1,FUNCT_1:def 6;
    p in B by A3,ZFMISC_1:32;
    then f.p in f.:B by A1,FUNCT_1:def 6;
    then {f.p,f.q} c= f.:B by A4,ZFMISC_1:32;
    hence thesis by PENCIL_1:def 1;
  end;
end;
