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 Th8:
  q in M & q in N & a 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 & q=a9
  implies q=b9
proof
  assume that
A1: q in M and
A2: q in N and
A3: a in M and
A4: b in N and
A5: b9 in N and
A6: q<>a and
A7: q<>b & M<>N and
A8: a,b // a9,b9 or b,a // b9,a9 and
A9: M is being_line and
A10: N is being_line and
A11: q=a9;
A12: not LIN q,a,b
  proof
    assume not thesis;
    then consider A such that
A13: A is being_line & q in A and
A14: a in A and
A15: b in A by AFF_1:21;
    M=A by A1,A3,A6,A9,A13,A14,AFF_1:18;
    hence contradiction by A2,A4,A7,A10,A13,A15,AFF_1:18;
  end;
  LIN q,b,b9 & a,b // a9,b9 by A2,A4,A5,A8,A10,AFF_1:4,21;
  hence thesis by A11,A12,AFF_1:55;
end;
