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 Th14:
  for n,m,r being Integer holds
  k <> i implies divisors(n,m,k) misses divisors(n,m,i)
  proof
  let n,m,r be Integer;
  assume
A1: k<>i;
    assume divisors(n,m,k) meets divisors(n,m,i);
    then consider x be object such that
A2: x in divisors(n,m,k) & x in divisors(n,m,i) by XBOOLE_0:3;
    reconsider x as Nat by A2;
    x mod m = k & x mod m=i by A2,Th12;
    hence thesis by A1;
  end;
