theorem Th93:
  f is collineation implies f.:(Line(a,b))=Line(f.a,f.b)
proof
  assume
A1: f is collineation;
  now
    let x be object;
    assume
A2: x in Line(f.a,f.b);
    then reconsider x9=x as Element of AFS;
    consider y such that
A3: f.y=x9 by Th1;
    LIN f.a,f.b,f.y by A2,A3,AFF_1:def 2;
    then LIN a,b,y by A1,Th88;
    then y in Line(a,b) by AFF_1:def 2;
    hence x in f.:(Line(a,b)) by A3,Th91;
  end;
  then
A4: Line(f.a,f.b) c= f.:(Line(a,b)) by TARSKI:def 3;
  now
    let x be object;
    assume
A5: x in f.:(Line(a,b));
    then reconsider x9=x as Element of AFS;
    consider y such that
A6: y in Line(a,b) and
A7: f.y=x9 by A5,Th91;
    LIN a,b,y by A6,AFF_1:def 2;
    then LIN f.a,f.b,x9 by A1,A7,Th88;
    hence x in Line(f.a,f.b) by AFF_1:def 2;
  end;
  then f.:(Line(a,b)) c= Line(f.a,f.b) by TARSKI:def 3;
  hence thesis by A4,XBOOLE_0:def 10;
end;
