reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem
  for a,b,c,n being Integer st 3 < n
   ex k being Integer st
    not n divides k+a & not n divides k+b & not n divides k+c
  proof
    let a,b,c,n be Integer such that
A1: 3 < n;
    set r1 = (-a) mod n;
    set r2 = (-b) mod n;
    set r3 = (-c) mod n;
    reconsider X = n as Nat by A1,TARSKI:1;
    3+1 <= n by A1,NAT_1:13;
    then card Segm 4 c= card Segm X by NAT_1:40;
    then consider r being object such that
A2: r in X and
A3: r1 <> r & r2 <> r & r3 <> r by GLIBPRE0:17;
    reconsider r as Nat by A2;
    take r;
    Segm X = X;
    then r < n by A2,NAT_1:44;
    hence thesis by A3,Lm38;
  end;
