reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem Th1:
  (LIN p,a,a9 or LIN p,a9,a) & p<>a implies ex b9 st LIN p,b,b9 & a ,b // a9,b9
proof
  assume that
A1: LIN p,a,a9 or LIN p,a9,a and
A2: p<>a;
  LIN p,a,a9 by A1,AFF_1:6;
  then p,a // p,a9 by AFF_1:def 1;
  then a,p // p,a9 by AFF_1:4;
  then consider b9 such that
A3: b,p // p,b9 and
A4: b,a // a9,b9 by A2,DIRAF:40;
  p,b // p,b9 by A3,AFF_1:4;
  then
A5: LIN p,b, b9 by AFF_1:def 1;
  a,b // a9,b9 by A4,AFF_1:4;
  hence thesis by A5;
end;
