let a, c, b be real number ; :: thesis: ( a > 0 & a <= 1 & c >= b implies a #R c <= a #R b )
assume that
A1: a > 0 and
A2: a <= 1 and
A3: c >= b ; :: thesis: a #R c <= a #R b
1 / a >= 1 by A1, A2, Lm4, XREAL_1:87;
then (1 / a) #R c >= (1 / a) #R b by A3, Th96;
then 1 / (a #R c) >= (1 / a) #R b by A1, Th93;
then A4: 1 / (a #R c) >= 1 / (a #R b) by A1, Th93;
a #R b > 0 by A1, Th95;
hence a #R c <= a #R b by A4, XREAL_1:91; :: thesis: verum