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;

theorem Th35:
  Gen w,y implies for u,v being VECTOR of V st u<>v holds PProJ(w,
  y,u-v,u-v) <> 0
proof
  assume
A1: Gen w,y;
  let u,v be VECTOR of V;
  assume
A2: u<>v;
  assume PProJ(w,y,u-v,u-v) = 0;
  then u-v,u-v are_Ort_wrt w,y by A1,Th32;
  then u-v = 0.V by A1,ANALMETR:11;
  hence contradiction by A2,RLVECT_1:21;
end;
