
theorem Th2:
  for X being set, Y being non empty set, p being FinSequence of X
  holds (X --> Y)#.p = (len p)-tuples_on Y
proof
  let X be set, Y be non empty set, p be FinSequence of X;
  rng p c= X by FINSEQ_1:def 4;
  then rng p /\ X = rng p by XBOOLE_1:28;
  then
A1: p"X = p"rng p by RELAT_1:133
    .= dom p by RELAT_1:134;
  p in X* by FINSEQ_1:def 11;
  hence (X --> Y)#.p = product ((X --> Y)*p) by FINSEQ_2:def 5
    .= product (p"X --> Y) by FUNCOP_1:19
    .= product ((Seg len p) --> Y) by A1,FINSEQ_1:def 3
    .= (len p)-tuples_on Y by Lm1;
end;
