reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r for Real;
reserve p,p1,p2 for Prime;

theorem Th5:
  i divides j implies i,j+1 are_coprime
  proof
    assume
A1: i divides j;
A2: 1 divides i & 1 divides j+1 by INT_2:12;
    for m being Integer st m divides i & m divides j+1 holds m divides 1
    proof
      let m be Integer;
      assume m divides i;
      then
A3:   m divides j by A1,INT_2:9;
      assume m divides j+1;
      then m divides j+1-j by A3,INT_5:1;
      hence thesis;
    end;
    hence i gcd (j+1) = 1 by A2,INT_2:def 2;
  end;
