
theorem
  for V being RealUnitarySpace, v,w being VECTOR of V, x being set holds
  x in v + Lin{w} iff ex a being Real st x = v + a * w
proof
  let V be RealUnitarySpace;
  let v,w be VECTOR of V;
  let x be set;
  thus x in v + Lin{w} implies ex a being Real st x = v + a * w
  proof
    assume x in v + Lin{w};
    then consider u being VECTOR of V such that
A1: u in Lin{w} and
A2: x = v + u by RUSUB_2:63;
    ex a being Real st u = a * w by A1,Th29;
    hence thesis by A2;
  end;
  given a be Real such that
A3: x = v + a * w;
  a * w in Lin{w} by Th29;
  hence thesis by A3,RUSUB_2:63;
end;
