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 Th27:
  Gen w,y implies for p,q,r ex r1 st p,q _|_ r,r1 & r<>r1
proof
  assume
A1: Gen w,y;
  let p,q,r;
  reconsider u=p,v=q,u1=r as Element of V;
  consider v2 such that
A2: v-u,v2 are_Ort_wrt w,y and
A3: v2<>0.V by A1,Th8;
  set v1 = u1+v2;
  reconsider r1=v1 as Element of AMSpace(V,w,y);
A4: v1-u1 = v2+(u1-u1) by RLVECT_1:def 3
    .= v2+0.V by RLVECT_1:15
    .= v2 by RLVECT_1:4;
  then u,v,u1,v1 are_Ort_wrt w,y by A2;
  then
A5: p,q _|_ r,r1 by Th21;
  r<>r1 by A3,A4,RLVECT_1:15;
  hence thesis by A5;
end;
