
theorem Th7:
  for V being RealUnitarySpace, W being Subspace of V, v being
  VECTOR of V, w being VECTOR of W, a being Real st w = v
    holds a * w = a * v
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let v be VECTOR of V;
  let w be VECTOR of W;
  let a be Real;
  assume
A1: w = v;
  reconsider aa=a as Element of REAL by XREAL_0:def 1;
  aa * w = ((the Mult of V) | [:REAL, the carrier of W:]).[aa,w] by Def1;
  hence thesis by A1,FUNCT_1:49;
end;
