theorem
  M _|_ K & N _|_ K implies M // N
proof
  assume that
A1: M _|_ K and
A2: N _|_ K;
  consider p1,q1,a,b being Element of POS such that
A3: p1<>q1 and
A4: a<>b and
A5: K = Line(p1,q1) and
A6: M = Line(a,b) and
A7: p1,q1 _|_ a,b by A1,Th45;
  consider p2,q2,c,d being Element of POS such that
A8: p2<>q2 and
A9: c <>d and
A10: K = Line(p2,q2) and
A11: N = Line(c,d) and
A12: p2,q2 _|_ c,d by A2,Th45;
  reconsider p19=p1,p29=p2,q19=q1,q29=q2 as Element of the AffinStruct of POS;
A13: Line(p19,q19) = Line(p2,q2) by A5,A10,Th41
    .= Line(p29,q29) by Th41;
  then q29 in Line(p19,q19) by AFF_1:15;
  then
A14: LIN p19,q19,q29 by AFF_1:def 2;
  p29 in Line(p19,q19) by A13,AFF_1:15;
  then LIN p19,q19,p29 by AFF_1:def 2;
  then p19,q19 // p29,q29 by A14,AFF_1:10;
  then p1,q1 // p2,q2 by Th36;
  then a,b _|_ p2,q2 by A3,A7,Th62;
  then a,b // c,d by A8,A12,Th63;
  hence thesis by A4,A6,A9,A11;
end;
