reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;
reserve f for FinSequence;
reserve x,X,Y for set;
reserve f1,f2,f3 for FinSequence of REAL;
reserve n1,n2,m1,m2 for Nat;

theorem Th14:
  n1 in NatDivisors n & m1 in NatDivisors m & n,m
  are_coprime implies n1,m1 are_coprime
proof
A1: n1 gcd m1 divides m1 by INT_2:def 2;
  assume n1 in NatDivisors n;
  then
A2: n1 divides n by MOEBIUS1:39;
  n1 gcd m1 divides n1 by INT_2:def 2;
  then
A3: n1 gcd m1 divides n by A2,INT_2:9;
  assume
A4: m1 in NatDivisors m;
  then m1 divides m by MOEBIUS1:39;
  then
A5: n1 gcd m1 divides m by A1,INT_2:9;
  0 < m1 by A4,MOEBIUS1:39;
  then n1 gcd m1 <> 0 by INT_2:5;
  then n1 gcd m1 + 1 > 0 + 1 by XREAL_1:8;
  then
A6: n1 gcd m1 >= 1 by NAT_1:13;
  assume
A7: n,m are_coprime;
  assume not n1,m1 are_coprime;
  then
A8: n1 gcd m1 <> 1 by INT_2:def 3;
  n gcd m > 0 by A7,INT_2:def 3;
  then n1 gcd m1 <= n gcd m by A3,A5,INT_2:22,27;
  then n gcd m <> 1 by A8,A6,XXREAL_0:1;
  hence contradiction by A7,INT_2:def 3;
end;
