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
  n>1 & m>1 & m,n are_coprime & i,m*n are_coprime
    implies order(i,m*n) = order(i,m) lcm order(i,n)
proof
assume
A1: n>1 & m>1 & m,n are_coprime & i,m*n are_coprime;
    then A2:m*n>1*1 by XREAL_1:98;
    then i <> 0 by A1,Th5; then
A3: i>0;
    set K = order(i,m) lcm order(i,n);
A4: m divides m*n & n divides m*n by NAT_D:def 3;then
    order(i,m) divides order(i,m*n) & order(i,n) divides order(i,m*n)
                                                       by A1,A2,Th20;
    then A5:K divides order(i,m*n) by NAT_D:def 4;
    i gcd m*n = 1 by A1,INT_2:def 3;
    then i gcd m = 1 & i gcd n = 1 by A4,WSIERP_1:16;then
A6: i,m are_coprime & i,n are_coprime by INT_2:def 3;
    i|^K-1>=0 by NAT_1:14,XREAL_1:48,A3;
    then A7:i|^K-1 is Element of NAT by INT_1:3;
    order(i,m) divides K & order(i,n) divides K by NAT_D:def 4;
    then (i|^K) mod m = 1 & (i|^K) mod n = 1 by A1,A6,PEPIN:48;
    then (i|^K),1 are_congruent_mod m & (i|^K),1 are_congruent_mod n
                                                      by A1,PEPIN:39;
    then m divides ((i|^K)-1) & n divides ((i|^K)-1) by INT_2:15;
    then m*n divides ((i|^K)-1) by A1,A7,PEPIN:4;
    then i|^K,1 are_congruent_mod m*n by INT_2:15;then
A8:i|^K mod m*n = 1 by A2,PEPIN:39;
    order(i,m*n) divides K by A1,A2,A8,PEPIN:47;
  hence thesis by A5,NAT_D:5;
end;
