reserve AFP for AffinPlane;
reserve a,a9,b,b9,c,c9,d,d9,o,p,p9,q,q9,r,s,t,x,y,z for Element of AFP;
reserve A,A9,C,D,P,B9,M,N,K for Subset of AFP;
reserve f for Permutation of the carrier of AFP;

theorem Th3:
  AFP is Desarguesian implies for o,a,b st o<>a & o<>b & LIN o,a,b
  ex f st f is dilatation & f.o=o & f.a=b
proof
  assume
A1: AFP is Desarguesian;
  let o,a,b;
  assume that
A2: o<>a and
A3: o<>b and
A4: LIN o,a,b;
  consider f such that
A5: for x,y holds (f.x=y iff ( not LIN o,a,x & LIN o,x,y & a,x // b,y or
(LIN o,a,x & ex p,p9 st not LIN o,a,p & LIN o,p,p9 & a,p // b,p9 & p,x // p9,y
  & LIN o,a,y))) by A1,A2,A3,A4,Lm13;
A6: f.a=b by A1,A2,A4,A5,Lm17;
A7: f.o=o by A1,A2,A4,A5,Lm17;
  f is dilatation by A1,A2,A4,A5,Lm17;
  hence thesis by A7,A6;
end;
