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 Th45:
  (a <> 0 or b <> 0) & n > 0 & a divides b|^n - 1 implies a,b are_coprime
  proof
    set g = a gcd b;
    assume (a <> 0 or b <> 0) & n > 0 & a divides b|^n - 1;
    then g divides b|^n & g divides b|^n - 1 by INT_2:2,21,NEWTON02:14;
    then g divides b|^n - (b|^n - 1) by INT_5:1;
    hence thesis by WSIERP_1:15;
  end;
