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 Th25:
  Gen w,y implies for p,q,r holds p,q _|_ r,r
proof
  assume
A1: Gen w,y;
  let p,q,r;
  reconsider u=p,v=q,u1=r as Element of V;
  u1-u1 = 0.V by RLVECT_1:15;
  then v-u,u1-u1 are_Ort_wrt w,y by A1,Th5;
  then u,v,u1,u1 are_Ort_wrt w,y;
  hence thesis by Th21;
end;
