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 Th22:
  modSeq(m,0) = NAT --> 0
proof
  set fm = modSeq(m,0);
A1: for x being object st x in dom fm holds fm.x = g.x
  proof
    defpred P[Nat] means fm.$1 = 0;
    let x be object;
    assume x in dom fm;
    then reconsider x as Element of NAT;
    fm.0 = m mod 0 by Def1
      .= 0;
    then
A2: P[0];
A3: for n being Nat holds P[n] implies P[n+1] by Th14,NAT_1:11;
    for n being Nat holds P[n] from NAT_1:sch 2(A2,A3);
    then fm.x = 0;
    hence thesis by FUNCOP_1:7;
  end;
  dom fm = NAT by FUNCT_2:def 1;
  hence thesis by A1;
end;
