
theorem Th3:
  for r being Real, v being VECTOR of Linear_Space_of_RealSequences holds
  r * v = r (#) seq_id(v)
proof
  let r be Real;
  let v be VECTOR of Linear_Space_of_RealSequences;
  reconsider r1 = r as Element of REAL by XREAL_0:def 1;
  reconsider v1 = v as Element of Funcs(NAT,REAL);
  reconsider h = (RealFuncExtMult NAT).(r1,v1) as Real_Sequence by FUNCT_2:66;
  h = r(#)seq_id(v)
  proof
    let n be Element of NAT;
    thus h.n = r*(v1.n) by FUNCSDOM:4
    .= (r(#)seq_id(v)).n by VALUED_1:6;
  end;
  hence thesis;
end;
