reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th13:
  for n,m,r,p being Integer holds
  k in divisors(n,m,r) \/ divisors(n,m,p)
   iff (k mod m = r or k mod m = p) & k divides n
   proof
     let n,m,r,p be Integer;
     hereby
       assume k in divisors(n,m,r)\/divisors(n,m,p);
       then k in divisors(n,m,r) or k in divisors(n,m,p) by XBOOLE_0:def 3;
       hence (k mod m = r or k mod m = p) & k divides n by Th12;
     end;
     assume (k mod m = r or k mod m = p) & k divides n;
     then k in divisors(n,m,r) or k in divisors(n,m,p);
     hence thesis by XBOOLE_0:def 3;
   end;
