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 Th21:
  for a,r being Real holds
  multreal.:( Seg 1 --> a, <* r *> ) = <* a * r *>
  proof
    let a,r be Real;
    reconsider r1 = a, r2 = r as Element of REAL by XREAL_0:def 1;
    a is Element of REAL by XREAL_0:def 1;
    then multreal.:( Seg 1 --> a, <* r *> ) = multreal.:(<* a *>, <* r *>)
                                             by Th20
                                           .= <* multreal.(r1,r2) *>
                                             by FINSEQ_2:74
                                           .= <* r1 * r2 *> by BINOP_2:def 11;
    hence thesis;
  end;
