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 Th33:
  Gen w,y implies for u,v,u1,v1 being VECTOR of V holds (u,v,u1,v1
  are_Ort_wrt w,y iff PProJ (w,y,v-u,v1-u1)=0)
proof
  assume
A1: Gen w,y;
  let u,v,u1,v1 be VECTOR of V;
  u,v,u1,v1 are_Ort_wrt w,y iff v-u,v1-u1 are_Ort_wrt w,y by ANALMETR:def 3;
  hence thesis by A1,Th32;
end;
