reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;
reserve V             for RealLinearSpace,
        A,B,C,P,Q,R,S for Element of V;

theorem Th27:
  for x being Element of TOP-REAL 1 for a,r being Real st
  x = <* a *> holds r * x = <* (r * a) *>
  proof
    let x be Element of TOP-REAL 1;
    let a,r be Real;
    assume x = <* a *>; then
A1: x.1 = a;
    reconsider x9 = x as Element of REAL 1 by EUCLID:22;
A2: (r * x9).1 = r * x.1 by RVSUM_1:44;
    r * x9 in REAL 1;
    then r * x is Element of TOP-REAL 1 by EUCLID:22;
    then consider b be Real such that
A3: r * x = <* b *> by JORDAN2B:20;
    thus thesis by A1,A3,A2;
  end;
