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 Th6:
  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;
      then m = 1 or m = -1 by INT_2:13;
      hence thesis by INT_2:14;
    end;
    hence i gcd (j-1) = 1 by A2,INT_2:def 2;
  end;
