
theorem Th25:
  for V being RealLinearSpace, A being Affine Subset of V st 0.V in A
  for x being VECTOR of V, a being Real st x in A holds a * x in A
proof
  let V be RealLinearSpace;
  let A be Affine Subset of V;
  assume
A1: 0.V in A;
  for x being VECTOR of V, a being Real st x in A holds a * x in A
  proof
    let x be VECTOR of V;
    let a be Real;
    assume x in A;
    then (1-a) * 0.V + a * x in A by A1,Def4;
    then 0.V + a * x in A;
    hence thesis;
  end;
  hence thesis;
end;
