let b, d, a, c be real number ; :: thesis: ( b < 0 & d < 0 & a * d < c * b implies a / b < c / d )
assume that
A1: b < 0 and
A2: d < 0 and
A3: a * d < c * b ; :: thesis: a / b < c / d
(a * d) / b > c by A1, A3, Th86;
then (a / b) * d > c by XCMPLX_1:74;
hence a / b < c / d by A2, Th86; :: thesis: verum