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
  a in K & b in K & c,d _|_ K implies c,d _|_ a,b & a,b _|_ c,d
proof
  assume that
A1: a in K and
A2: b in K and
A3: c,d _|_ K;
  consider p,q such that
A4: p<>q and
A5: K = Line(p,q) and
A6: c,d _|_ p,q by A3;
  reconsider a9=a,b9=b, p9=p,q9=q as Element of the AffinStruct of POS;
  LIN p,q, b by A2,A5,Def10;
  then
A7: LIN p9,q9,b9 by Th40;
  LIN p,q,a by A1,A5,Def10;
  then LIN p9,q9,a9 by Th40;
  then p9,q9 // a9, b9 by A7,AFF_1:10;
  then
A8: p,q // a,b by Th36;
  p,q _|_ c,d by A6,Def7;
  hence c,d _|_ a,b by A4,A8,Def7;
  hence thesis by Def7;
end;
