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 Th26:
  p,p1 _|_ q,q1 & p,p1 // r,r1 implies p=p1 or q,q1 _|_ r,r1
proof
  assume that
A1: p,p1 _|_ q,q1 and
A2: p,p1 // r,r1;
  reconsider u=p,v=p1,u1=q,v1=q1,u2=r,v2=r1 as Element of V;
  consider a,b such that
A3: a*(v-u) = b*(v2-u2) and
A4: a<>0 or b<>0 by A2,Th22;
  assume
A5: p<>p1;
  b<>0
  proof
    assume
A6: b=0;
    then a*(v-u) = 0.V by A3,RLVECT_1:10;
    then v-u = 0.V by A4,A6,RLVECT_1:11;
    hence contradiction by A5,RLVECT_1:21;
  end;
  then
A7: v2-u2 = b"*(a*(v-u)) by A3,ANALOAF:5
    .= (b"*a)*(v-u) by RLVECT_1:def 7;
  u,v,u1,v1 are_Ort_wrt w,y by A1,Th21;
  then v-u,v1-u1 are_Ort_wrt w,y;
  then v2-u2,v1-u1 are_Ort_wrt w,y by A7,Th7;
  then v1-u1,v2-u2 are_Ort_wrt w, y;
  then u1,v1,u2,v2 are_Ort_wrt w,y;
  hence thesis by Th21;
end;
