
theorem
  for V being RealUnitarySpace, W being Subspace of V, v1,v2 being
VECTOR of V, w1,w2 being VECTOR of W st w1 = v1 & w2 = v2 holds w1 .|. w2 = v1
  .|. v2
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let v1,v2 be VECTOR of V;
  let w1,w2 be VECTOR of W;
  reconsider ww1 = w1, ww2 = w2 as VECTOR of V by Th3;
  assume w1 = v1 & w2 = v2;
  then
A1: v1 .|. v2 = (the scalar of V).[ww1,ww2];
  w1 .|. w2 = (the scalar of W).[w1,w2]
    .= ((the scalar of V)||the carrier of W).[w1,w2] by Def1;
  hence thesis by A1,FUNCT_1:49;
end;
