reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;

theorem Th8:
  q=i1/m1 & m1<>0 & i1,m1 are_coprime implies
    i1=numerator(q) & m1=denominator(q)
   proof
     assume that
A1:  q=i1/m1 and
A2:  m1<>0 and
A3:  i1,m1 are_coprime;
A4:  i1 gcd m1 = 1 by A3,INT_2:def 3;
     ex m be Nat st i1=numerator(q)*m & m1=denominator(q)*m
     by A1,A2,RAT_1:27; then
     consider m be Nat such that
A5:  i1 = m * numerator(q) and
A6:  m1 = m * denominator(q);
A7:  m divides i1 by A5;
A8:  m divides m1 by A6;
     m divides 1 by A4,A7,A8,INT_2:def 2;
     then m=1 by WSIERP_1:15;
     hence thesis by A6,A5;
   end;
