reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));
reserve POS for non empty ParOrtStr;
reserve p,p1,p2,q,q1,r,r1,r2 for Element of AMSpace(V,w,y);
reserve x,a,b,c,d,p,q,y for Element of POS;
reserve A,K,M for Subset of POS;
reserve POS for OrtAfSp;
reserve A,K,M,N for Subset of POS;
reserve a,b,c,d,p,q,r,s for Element of POS;

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;
