theorem
  f is collineation & K is being_line implies f.:K is being_line
proof
  assume that
A1: f is collineation and
A2: K is being_line;
  consider a,b such that
A3: a<>b and
A4: K=Line(a,b) by A2,AFF_1:def 3;
  set q=f.b;
  set p=f.a;
A5: p<>q by A3,FUNCT_2:58;
  f.:K=Line(p,q) by A1,A4,Th93;
  hence thesis by A5,AFF_1:def 3;
end;
