reserve a, b, k, n, m for Nat,
  i for Integer,
  r for Real,
  p for Rational,
  c for Complex,
  x for object,
  f for Function;

theorem Th21:
  divSeq(m,0) = NAT --> 0
proof
  set g = NAT --> 0;
  set fd = divSeq(m,0);
A1: for x being object st x in dom fd holds fd.x = g.x
  proof
    defpred P[Nat] means fd.$1 = 0;
    let x be object;
    assume x in dom fd;
    then reconsider x as Element of NAT;
A2: for n being Nat holds P[n] implies P[n+1]
    proof
      let n be Nat;
      assume
A3:   P[n];
      per cases;
      suppose
A4:     n = 0;
        fd.1 = 0 div (m mod 0) by Def2
          .= 0;
        hence thesis by A4;
      end;
      suppose
        0 <> n;
        hence thesis by A3,Th17,NAT_1:11;
      end;
    end;
    fd.0 = m div 0 by Def2
      .= 0;
    then
A5: P[0];
    for n being Nat holds P[n] from NAT_1:sch 2(A5,A2);
    then fd.x = 0;
    hence thesis by FUNCOP_1:7;
  end;
  dom fd = NAT by FUNCT_2:def 1;
  hence thesis by A1;
end;
