theorem Th51:
  a in K & b in K & a,b _|_ K implies a=b
proof
  assume that
A1: a in K and
A2: b in K and
A3: a,b _|_ K;
  consider p,q such that
A4: p<>q and
A5: K = Line(p,q) and
A6: a,b _|_ p,q by A3;
  reconsider a9=a,b9=b,p9=p,q9=q as Element of the AffinStruct of POS;
  set K9 = Line(p9,q9);
  b9 in K9 by A2,A5,Th41;
  then
A7: LIN p9,q9,b9 by AFF_1:def 2;
  a9 in K9 by A1,A5,Th41;
  then LIN p9,q9,a9 by AFF_1:def 2;
  then p9,q9 // a9,b9 by A7,AFF_1:10;
  then
A8: p,q // a,b by Th36;
  p,q _|_ a,b by A6,Def7;
  then a,b _|_ a,b by A4,A8,Def7;
  hence thesis by Def7;
end;
