
theorem Th30:
  for K be Field, V be non trivial VectSp of K for v,w be Vector
  of V for W be Linear_Compl of Lin{v} st v <> 0.V & w in W holds (
  coeffFunctional(v,W)).w = 0.K
proof
  let K be Field, V be non trivial VectSp of K, v,w be Vector of V, W be
  Linear_Compl of Lin{v};
  assume that
A1: v <> 0.V and
A2: w in W;
  set cf = coeffFunctional(v,W), cw = the carrier of W;
  reconsider s = w as Vector of W by A2;
  w in cw by A2;
  hence cf.(w) = (cf|cw).w by FUNCT_1:49
    .= (0Functional(W)).s by A1,Def8
    .= 0.K by HAHNBAN1:14;
end;
