reserve AS for AffinSpace,
  a,b,c,d,p,q,r,s,x for Element of AS;
reserve AFP for AffinPlane,
  a,a9,b,b9,c,c9,d,p,p9,q,q9,r,x,x9,y,y9,z for Element of AFP,
  A,C,P for Subset of AFP,
  f,g,h,f1,f2 for Permutation of the carrier of AFP;

theorem Th6:
  (for p,q,r st p<>q & LIN p,q,r holds r = p or r = q) & a,b // p,q
  & a,p // b,q & not LIN a,b,p implies a,q // b,p
proof
  assume that
A1: for p,q,r st p<>q & LIN p,q,r holds r = p or r = q and
A2: a,b // p,q and
A3: a,p // b,q and
A4: not LIN a,b,p;
  consider z such that
A5: q,p // a,z and
A6: q,a // p,z by DIRAF:40;
A7: p<>q
  proof
    assume p=q;
    then p,a // p,b by A3,AFF_1:4;
    then LIN p,a,b by AFF_1:def 1;
    hence contradiction by A4,AFF_1:6;
  end;
A8: not LIN a,p,q
  proof
A9: LIN p,q,p by AFF_1:7;
    assume LIN a,p,q;
    then
A10: LIN p,q,a by AFF_1:6;
    p,q // a,b by A2,AFF_1:4;
    then LIN p,q,b by A7,A10,AFF_1:9;
    hence contradiction by A4,A7,A10,A9,AFF_1:8;
  end;
A11: now
    assume a=z;
    then a,p // a,q by A6,AFF_1:4;
    hence contradiction by A8,AFF_1:def 1;
  end;
  p,q // a,z by A5,AFF_1:4;
  then a,b // a,z by A2,A7,AFF_1:5;
  then
A12: LIN a,b,z by AFF_1:def 1;
  a<>b by A4,AFF_1:7;
  then a=z or b=z by A1,A12;
  hence thesis by A6,A11,AFF_1:4;
end;
