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 Th32:
  Gen w,y implies for u,v being VECTOR of V holds (u,v are_Ort_wrt
  w,y iff PProJ(w,y,u,v)=0)
proof
  assume
A1: Gen w,y;
  let u,v be VECTOR of V;
  set a1=pr1(w,y,u),a2=pr2(w,y,u),b1=pr1(w,y,v),b2=pr2(w,y,v);
  u = a1*w+a2*y & v = b1*w+b2*y by A1,Lm15;
  hence thesis by A1,ANALMETR:1,def 2;
end;
