reserve n for Nat,
        i,j,i1,i2,i3,i4,i5,i6 for Element of n,
        p,q,r for n-element XFinSequence of NAT;
reserve i,j,n,n1,n2,m,k,l,u,e,p,t for Nat,
        a,b for non trivial Nat,
        x,y for Integer,
        r,q for Real;

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;
