let b, d, a, c be real number ; :: thesis: ( 0 < b & 0 < d & a * d < c * b implies a / b < c / d )
assume A1: ( b > 0 & d > 0 & a * d < c * b ) ; :: thesis: a / b < c / d
then (a * d) / b < c by Th85;
then (a / b) * d < c by XCMPLX_1:75;
hence a / b < c / d by A1, Th83; :: thesis: verum