reserve i,s,t,m,n,k for Nat,
        c,d,e for Element of NAT,
        fn for FinSequence of NAT,
        x,y for Integer;
reserve p for Prime;
 reserve fp,fr for FinSequence of NAT;

theorem Th20:
  n>1 & m>1 & i,n are_coprime & m divides n
    implies order(i,m) divides order(i,n)
proof
   assume A1:n>1 & m>1 & i,n are_coprime & m divides n;
   i gcd n = 1 by A1,INT_2:def 3;
   then i gcd m = 1 by A1,WSIERP_1:16;
   then A2:i,m are_coprime by INT_2:def 3;
   (i|^order(i,n)) mod n = 1 by A1,PEPIN:def 2
                        .= 1 mod n by A1,PEPIN:5;
   then i|^order(i,n),1 are_congruent_mod n by A1,NAT_D:64;
   then n divides (i|^order(i,n) - 1) by INT_2:15;
   then m divides (i|^order(i,n) - 1) by A1,INT_2:9;
   then i|^order(i,n),1 are_congruent_mod m by INT_2:15;
   then (i|^order(i,n)) mod m = 1 mod m by NAT_D:64
                               .= 1 by A1,PEPIN:5;
   hence thesis by A1,A2,PEPIN:47;
end;
