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 Th29:
  Gen w,y & p,q _|_ r,r1 & p,q _|_ r,r2 implies p,q _|_ r1,r2
proof
  assume that
A1: Gen w,y and
A2: p,q _|_ r,r1 and
A3: p,q _|_ r,r2;
  reconsider u=p,v=q,w1=r,v1=r1,v2=r2 as Element of V;
  u,v,w1,v2 are_Ort_wrt w,y by A3,Th21;
  then
A4: v-u,v2-w1 are_Ort_wrt w,y;
A5: (v2-w1)-(v1-w1) = v2-((v1-w1)+w1) by RLVECT_1:27
    .= v2-(v1-(w1-w1)) by RLVECT_1:29
    .= v2-(v1-0.V) by RLVECT_1:15
    .= v2-v1 by RLVECT_1:13;
  u,v,w1,v1 are_Ort_wrt w,y by A2,Th21;
  then v-u,v1-w1 are_Ort_wrt w,y;
  then v-u,(v2-w1)-(v1-w1) are_Ort_wrt w,y by A1,A4,Th10;
  then u,v,v1,v2 are_Ort_wrt w,y by A5;
  hence thesis by Th21;
end;
