reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;
reserve A,C,D,K for Subset of AS;

theorem
  for A be being_line Subset of AS st a,b // A & c,d // A holds a,b // c,d
proof
  let A be being_line Subset of AS;
  assume that
A1: a,b // A and
A2: c,d // A;
  consider p,q such that
A3: p<>q and
A4: A=Line(p,q) and
A5: a,b // p,q by A1;
A6: q in A by A4,Th14;
  p in A by A4,Th14;
  then c,d // p,q by A2,A3,A6,Th26;
  hence thesis by A3,A5,Th4;
end;
