reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;

theorem Th9:
  g divides i & g divides m implies i / m = (i div g) / (m div g)
  proof
    assume that
A1: g divides i and
A2: g divides m;
    per cases;
    suppose g = 0;
      hence thesis by A1;
    end;
    suppose
A4:   m = 0;
      i / 0 = 0 & (i div g) / (0 div g) = 0 by XCMPLX_1:49;
      hence thesis by A4;
    end;
    suppose that
A5:   g <> 0 and
A6:   m <> 0;
      g <= m by A2,A6,INT_2:27;
      then
A7:   m div g <> 0 by A5,NAT_2:13;
      i mod g = 0 & m mod g = 0 by A1,A2,A5,INT_1:62;
      then m div g = m / g & i div g = i / g by PEPIN:63;
      then i * (m div g) = m * (i div g);
      hence thesis by A6,A7,XCMPLX_1:94;
    end;
  end;
