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);

theorem Th28:
  Gen w,y & p,p1 _|_ q,q1 & p,p1 _|_ r,r1 implies p=p1 or q,q1 // r,r1
proof
  assume that
A1: Gen w,y and
A2: p,p1 _|_ q,q1 and
A3: p,p1 _|_ r,r1;
  reconsider u=p,v=p1,u1=q,v1=q1,u2=r,v2=r1 as Element of V;
  u,v,u2,v2 are_Ort_wrt w,y by A3,Th21;
  then
A4: v-u,v2-u2 are_Ort_wrt w,y;
  assume p<>p1;
  then
A5: v-u <> 0.V by RLVECT_1:21;
  u,v,u1,v1 are_Ort_wrt w,y by A2,Th21;
  then v-u,v1-u1 are_Ort_wrt w,y;
  then ex a,b st a*(v1-u1) = b*(v2-u2) & (a<>0 or b<>0) by A1,A4,A5,Th9;
  hence thesis by Th22;
end;
