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
  Gen w,y & p,q _|_ p1,p2 & p1,q _|_ p2,p implies p2,q _|_ p,p1
proof
  assume that
A1: Gen w,y and
A2: p,q _|_ p1,p2 and
A3: p1,q _|_ p2,p;
  reconsider u=p,v=q,u1=p1,u2=p2 as Element of V;
  u,v,u1,u2 are_Ort_wrt w,y by A2,Th21;
  then
A4: v-u,u2-u1 are_Ort_wrt w,y;
  u1,v,u2,u are_Ort_wrt w,y by A3,Th21;
  then
A5: v-u1,u-u2 are_Ort_wrt w,y;
A6: now
    let u,v,w;
    thus (u-v)-(u-w) = (w-u) + (u-v) by RLVECT_1:33
      .= w-v by ANALOAF:1;
  end;
  then
A7: (v-u)-(v-u1)=u1-u;
  (v-u1)-(v-u2)=u2-u1 & (v-u2)-(v-u)=u-u2 by A6;
  then v-u2,(v-u)-(v-u1) are_Ort_wrt w,y by A1,A4,A5,Th12;
  then u2,v,u,u1 are_Ort_wrt w,y by A7;
  hence thesis by Th21;
end;
