reserve a,b,c,k,l,m,n for Nat,
  i,j,x,y for Integer;

theorem Th9:
  m,n are_coprime implies ex k st (ex i0,j0 being Integer
st k = i0*m + j0*n & k > 0) & for l st (ex i,j being Integer st l = i*m + j*n &
  l > 0) holds k <= l
proof
  defpred P[Nat] means ex i0,j0 being Integer st $1 = i0*m + j0*n & $1>0;
  assume
A1: m,n are_coprime;
  m > 0 or n > 0
  proof
    assume ( not m > 0)& not n > 0;
    then m = 0 & n = 0;
    then m gcd n <> 1 by NAT_D:32;
    hence contradiction by A1,INT_2:def 3;
  end;
  then 1*m + 1*n > 0;
  then
A2: ex k be Nat st P[k];
  consider k be Nat such that
A3: P[k] and
A4: for l be Nat st P[l] holds k<=l from NAT_1:sch 5(A2);
  take k;
  thus ex i0,j0 being Integer st k = i0*m + j0*n & k > 0 by A3;
  let l;
  thus thesis by A4;
end;
