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
  A // M & M '||' X implies A '||' X
proof
  assume that
A1: A // M and
A2: M '||' X;
A3: M is being_line by A1,AFF_1:36;
A4: A is being_line by A1,AFF_1:36;
  now
    consider q,p such that
A5: q in A and
    p in A and
    q<>p by A4,AFF_1:19;
    let a,C;
    assume that
A6: a in X and
A7: C is being_line & C c= A;
A8: a*A = a*(q*M) by A1,A3,A5,Def3
      .= a*M by A3,Th31;
    C = A by A4,A7,Th33;
    hence a*C c= X by A2,A3,A6,A8;
  end;
  hence thesis;
end;
