reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th18:
  a <> 0 or b <> 0 implies
  ex A,B being Nat st a = (a gcd b)*A & b = (a gcd b)*B & A,B are_coprime
  proof
    set d = a gcd b;
    assume
A1: a <> 0 or b <> 0;
    then consider A,B being Integer such that
A2: a = d*A & b = d*B and
A3: A,B are_coprime by INT_2:23;
    0 <= A & 0 <= B by A1,A2;
    then reconsider A,B as Element of NAT by INT_1:3;
    take A,B;
    thus thesis by A2,A3;
  end;
