let m, n be Nat; :: thesis: for i, j being Integer st m <> 0 & n <> 0 holds
( denominator ((i / m) * (j / n)) = (m * n) / ((i * j) gcd (m * n)) & numerator ((i / m) * (j / n)) = (i * j) / ((i * j) gcd (m * n)) )
let i, j be Integer; :: thesis: ( m <> 0 & n <> 0 implies ( denominator ((i / m) * (j / n)) = (m * n) / ((i * j) gcd (m * n)) & numerator ((i / m) * (j / n)) = (i * j) / ((i * j) gcd (m * n)) ) )
assume A1: ( m <> 0 & n <> 0 ) ; :: thesis: ( denominator ((i / m) * (j / n)) = (m * n) / ((i * j) gcd (m * n)) & numerator ((i / m) * (j / n)) = (i * j) / ((i * j) gcd (m * n)) )
hence denominator ((i / m) * (j / n)) = (m * n) div ((i * j) gcd (m * n)) by Th25
.= (m * n) / ((i * j) gcd (m * n)) by Th8 ;
:: thesis: numerator ((i / m) * (j / n)) = (i * j) / ((i * j) gcd (m * n))
thus numerator ((i / m) * (j / n)) = (i * j) div ((i * j) gcd (m * n)) by A1, Th25
.= (i * j) / ((i * j) gcd (m * n)) by Th7 ; :: thesis: verum