reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th1:
  for a being Element of D, m being non zero Nat, g being
  FinSequence of D holds (len g = m & for i be Nat st i in dom g holds g.i = a)
  iff g = m |-> a
proof
  let a be Element of D, m be non zero Nat, g be FinSequence of D;
  hereby
    assume that
A1: len g = m and
A2: for i be Nat st i in dom g holds g.i = a;
    dom g = dom(m |-> a) & for i be Nat st i in dom g holds g.i = (m |-> a ).i
    proof
      thus dom g = Seg m by A1,FINSEQ_1:def 3
        .= dom(m |-> a) by FUNCOP_1:13;
      let i be Nat such that
A3:   i in dom g;
A4:   i in Seg m by A1,A3,FINSEQ_1:def 3;
      thus g.i = a by A2,A3
        .= (m |-> a).i by A4,FINSEQ_2:57;
    end;
    hence g = m |-> a by FINSEQ_1:13;
  end;
  assume
A5: g = m |-> a;
  then dom g = Seg m by FUNCOP_1:13;
  hence thesis by A5,CARD_1:def 7,FINSEQ_2:57;
end;
