reserve a,b,m,x,n,l,xi,xj for Nat,
  t,z for Integer;

theorem
  a <> 0 & m > 1 & a,m are_coprime implies
  a |^ Euler m, 1 are_congruent_mod m
  proof
    assume that
A1: a <> 0 and
A2: m > 1 and
A3: a,m are_coprime;
    (a |^ Euler m) mod m = 1 by A1,A2,A3,Th18
    .= 1 mod m by A2,NAT_D:14;
    hence thesis by A2,NAT_D:64;
  end;
