reserve i,j,n,n1,n2,m,k,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat;

theorem Th29:
  a,b are_congruent_mod k implies Py(a,n),Py(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 = Py(a,n)-Py(b,n) by Th28,INT_1:def 5;
  p*(a-b) = p*(x*k) by A1
    .= p*x*k;
  hence thesis by A2, INT_1:def 5;
end;
