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
  M is being_line & N is being_line implies (M // N iff M '||' N)
proof
  assume that
A1: M is being_line and
A2: N is being_line;
A3: now
    assume
A4: M // N;
    now
      let a,A;
      assume that
A5:   a in N and
A6:   A is being_line & A c= M;
      N=a*M by A1,A4,A5,Def3;
      hence a*A c= N by A1,A6,Th33;
    end;
    hence M '||' N;
  end;
  now
    consider a,b such that
A7: a in N and
    b in N and
    a<>b by A2,AFF_1:19;
A8: a*M is being_line by A1,Th27;
    assume M '||' N;
    then a*M c= N by A1,A7;
    then a*M=N by A2,A8,Th33;
    hence M // N by A1,Def3;
  end;
  hence thesis by A3;
end;
