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 Th22:
  for a being Real for p being Tuple of 1,REAL holds
  multreal.:(dom p --> a,p) = a * p
  proof
    let a be Real;
    let p be Tuple of 1,REAL;
    consider d be Element of REAL such that
A1: p = <* d *> by FINSEQ_2:97;
A2: a * p = <* a * d *> by A1,RVSUM_1:47;
    dom p = Seg 1 by A1,FINSEQ_1:def 8;
    hence thesis by A1,A2,Th21;
  end;
