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 Th30:
  Gen w,y & p,q _|_ p,q implies p = q
proof
  assume that
A1: Gen w,y and
A2: p,q _|_ p,q;
  reconsider u=p,v=q as Element of V;
  u,v,u,v are_Ort_wrt w,y by A2,Th21;
  then v-u,v-u are_Ort_wrt w,y;
  then v-u = 0.V by A1,Th11;
  hence thesis by RLVECT_1:21;
end;
