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 Th10:
  (M // N or N // M) & a in M & b in N & b9 in N & M<>N & (a,b //
  a9,b9 or b,a // b9,a9) & a=a9 implies b=b9
proof
  assume that
A1: M // N or N // M and
A2: a in M and
A3: b in N & b9 in N and
A4: M<>N and
A5: ( a,b // a9,b9 or b,a // b9,a9)& a=a9;
  a,b // a,b9 by A5,AFF_1:4;
  then LIN a,b,b9 by AFF_1:def 1;
  then
A6: LIN b,b9,a by AFF_1:6;
  assume
A7: b<>b9;
  N is being_line by A1,AFF_1:36;
  then a in N by A3,A6,A7,AFF_1:25;
  hence contradiction by A1,A2,A4,AFF_1:45;
end;
