theorem Th23:
    Px(a,|.4*x*n+y.|), Px(a,|.y.|) are_congruent_mod Px(a,|.x.|)
proof
  defpred P[Nat] means  Px(a,|.4*x*$1+y.|), Px(a,|.y.|)
  are_congruent_mod Px(a,|.x.|);
A1: P[0] by INT_1:11;
A2: P[k] implies P[k+1]
  proof
    Px(a,|.4*x+(4*x*k+y).|), Px(a,|.4*x*k+y.|)
    are_congruent_mod Px(a,|.x.|)by Th18;
    hence thesis by INT_1:15;
  end;
  P[k] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
