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 Th16:
  a,b are_congruent_mod k implies Px(a,n),Px(b,n) are_congruent_mod k
proof
  assume a,b are_congruent_mod k;
  then consider x be Integer such that
  A1: k*x = a-b by INT_1:def 5;
  consider p be Integer such that
  A2: (a-b)*p = Px(a,n)-Px(b,n) by HILB10_1:25,INT_1:def 5;
  p*(a-b) = p*(x*k) by A1
         .= p*x*k;
  hence thesis by A2, INT_1:def 5;
end;
