
theorem Th12:
  for X be RealLinearSpace-Sequence,
      r be Element of REAL,
      v be Element of product carr X,
      i be Element of dom carr X
  holds
    ([:multop X:].(r,v)).i = (the Mult of X.i).(r,v.i)
  & for vi be VECTOR of X.i st vi = v.i
    holds ([:multop X:].(r,v)).i = r * vi
  proof
    let X be RealLinearSpace-Sequence;
    let r be Element of REAL,v be Element of product carr X;
    let i be Element of dom carr X;
    ([:multop X:].(r,v)).i = ((multop X).i).(r,v.i) by PRVECT_2:def 2;
    hence thesis by PRVECT_2:def 8;
  end;
