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 Th9:
  q in M & q in N & a in M & a9 in M & b in N & b9 in N & q<>a & q
<>b & M<>N & (a,b // a9,b9 or b,a // b9,a9) & M is being_line & N is being_line
  & a=a9 implies b=b9
proof
  assume that
A1: q in M and
A2: q in N and
A3: a in M and
A4: a9 in M and
A5: b in N and
A6: b9 in N and
A7: q<>a and
A8: q<>b & M<>N and
A9: a,b // a9,b9 or b,a // b9,a9 and
A10: M is being_line and
A11: N is being_line and
A12: a=a9;
A13: a,b // a9,b & a,b // a9,b9 by A9,A12,AFF_1:2,4;
A14: not LIN q,a,b
  proof
    assume not thesis;
    then consider A such that
A15: A is being_line & q in A and
A16: a in A and
A17: b in A by AFF_1:21;
    M=A by A1,A3,A7,A10,A15,A16,AFF_1:18;
    hence contradiction by A2,A5,A8,A11,A15,A17,AFF_1:18;
  end;
A18: LIN q,b,b by AFF_1:7;
  LIN q,a,a9 & LIN q,b,b9 by A1,A2,A3,A4,A5,A6,A10,A11,AFF_1:21;
  hence thesis by A14,A18,A13,AFF_1:56;
end;
