let a, c, b be real number ; :: thesis: ( a >= 1 & c >= b implies a #R c >= a #R b )
assume that
A1: a >= 1 and
A2: c >= b ; :: thesis: a #R c >= a #R b
consider s1 being Rational_Sequence such that
A3: s1 is convergent and
A4: c = lim s1 and
A5: for n being Element of NAT holds s1 . n >= c by Th80;
A6: a #Q s1 is convergent by A1, A3, Th82;
consider s2 being Rational_Sequence such that
A7: s2 is convergent and
A8: b = lim s2 and
A9: for n being Element of NAT holds s2 . n <= b by Th79;
A10: a #Q s2 is convergent by A1, A7, Th82;

now

let n be Element of NAT ; :: thesis: (a #Q s1) . n >= (a #Q s2) . n
s1 . n >= c by A5;
then A11: s1 . n >= b by A2, XXREAL_0:2;
s2 . n <= b by A9;
then s1 . n >= s2 . n by A11, XXREAL_0:2;
then a #Q (s1 . n) >= a #Q (s2 . n) by A1, Th74;
then a #Q (s1 . n) >= (a #Q s2) . n by Def7;
hence (a #Q s1) . n >= (a #Q s2) . n by Def7; :: thesis: verum

end;

then lim (a #Q s1) >= lim (a #Q s2) by A6, A10, SEQ_2:32;
then a #R c >= lim (a #Q s2) by A1, A3, A4, A6, Def8;
hence a #R c >= a #R b by A1, A7, A8, A10, Def8; :: thesis: verum