let a, b be real number ; :: thesis: ( 0 < a & a < b implies b " < a " )
A1: b " = 1 / b by XCMPLX_1:217;
assume that
A2: 0 < a and
A3: a < b ; :: thesis: b " < a "
a * (b " ) < b * (b " ) by A2, A3, Lm13;
then a * (b " ) < 1 by A1, A2, A3, XCMPLX_1:88;
then (a " ) * (a * (b " )) < 1 * (a " ) by A2, Lm13;
then ((a " ) * a) * (b " ) < 1 * (a " ) ;
then 1 * (b " ) < 1 * (a " ) by A2, XCMPLX_0:def 7;
hence b " < a " ; :: thesis: verum