
theorem Th2:
  for I being set,
      Y being non empty set,
      x being Y-valued ManySortedSet of I,
      p being FinSequence of I holds
  p*x is FinSequence of Y
  proof
    let I be set, Y be non empty set,
    x be Y-valued ManySortedSet of I,
    p be FinSequence of I;
A1: dom x =I by PARTFUN1:def 2;
A2: rng x c= Y by RELAT_1:def 19;
    then
A3: x is Function of I,Y by A1,FUNCT_2:2;
    reconsider x1=x as Function of I,Y by A1,A2,FUNCT_2:2;
    reconsider xp=p*x1 as FinSequence;
    thus thesis by A3,FINSEQ_2:32;
  end;
