reserve a,b,p,k,l,m,n,s,h,i,j,t,i1,i2 for natural Number;

theorem Th22:
  (p+s) mod n = ((p mod n)+s) mod n
proof
  per cases;
  suppose n > 0;
    then (p + s) mod n = (n*(p div n) + (p mod n) + s) mod n by INT_1:59
      .=(n*(p div n) + ((p mod n) + s)) mod n
      .=((p mod n) + s) mod n by Th21;
    hence thesis;
  end;
  suppose
A1: n = 0;
    hence (p+s) mod n = 0 by Def2
      .= ((p mod n)+s) mod n by A1,Def2;
  end;
end;
