
theorem Th5:
  for p be FinSequence, a be object holds
  p = (len p) |-> a
  iff (for k be object st k in dom p holds p.k = a)
  proof
    let p be FinSequence, a be object;
    thus p = (len p) |-> a implies
        for k be object st k in dom p holds p.k = a
    proof
      assume
      A2: p = (len p) |-> a;
      let k be object;
      assume k in dom p; then
      k in Seg len p by FINSEQ_1:def 3;
      hence thesis by A2,FINSEQ_2:57;
    end;
    assume for k be object st k in dom p holds p.k = a; then
    p = dom p --> a by FUNCOP_1:11;
    hence thesis by FINSEQ_1:def 3;
  end;
