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 & s,n are_coprime & t,n are_coprime &
  order(s*t,n) = order(s,n)*order(t,n) implies
  order(s,n),order(t,n) are_coprime
proof assume A1:n>1 & s,n are_coprime & t,n are_coprime &
   order(s*t,n) = order(s,n)*order(t,n);
   then s gcd n = 1 & t gcd n = 1 by INT_2:def 3; then
   s*t gcd n = 1 by WSIERP_1:7;
   then A2:s*t,n are_coprime by INT_2:def 3;
   set L = order(s,n) lcm order(t,n);
   order(s,n)>0 & order(t,n)>0 by A1,PEPIN:def 2;
   then A3:L<>0 by INT_2:4;
   order(s,n) divides order(s*t,n) & order(t,n) divides order(s*t,n)
                                                       by A1,NAT_D:def 3;
   then A4:L divides order(s*t,n) by NAT_D:def 4;
A5:order(s,n) divides L & order(t,n) divides L by NAT_D:def 4;then
   s|^(order(s,n) lcm order(t,n)) mod n = 1 by A1,PEPIN:48
                                       .= 1 mod n by A1,PEPIN:5;
then A6:s|^(order(s,n) lcm order(t,n)),1 are_congruent_mod n by A1,NAT_D:64;
   t|^L mod n = 1 by A1,A5,PEPIN:48   .= 1 mod n by A1,PEPIN:5;
   then t|^L,1 are_congruent_mod n by A1,NAT_D:64;
   then A7:t|^(order(s,n) lcm order(t,n))*s|^(order(s,n) lcm order(t,n)),1*1
          are_congruent_mod n by A6,INT_1:18;
   set B=s*t;
   B|^(order(s,n) lcm order(t,n)),1*1 are_congruent_mod n by A7,NEWTON:7;
   then A8:B|^(order(s,n) lcm order(t,n)) mod n = 1 mod n by NAT_D:64
                                              .= 1 by A1,PEPIN:5;
   order(B,n) divides L by A8,A1,A2,PEPIN:47;
   then L = order(s,n)*order(t,n) by A1,A4,NAT_D:5;
   then L*(order(s,n) gcd order(t,n)) = L*1 by NAT_D:29;
   then order(s,n) gcd order(t,n) = 1 by A3,XCMPLX_1:5;
   hence thesis by INT_2:def 3;
end;
