let a, b, c be Real; :: 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 p being Rational such that
A3: b < p and
A4: p < c by ;
consider q being Rational such that
A5: b < q and
A6: q < p by ;
consider s2 being Rational_Sequence such that
A7: s2 is convergent and
A8: b = lim s2 and
A9: for n being Nat holds s2 . n <= b by Th67;
A10: a #Q q < a #Q p by A1, A6, Th64;
now :: thesis: for n being Nat holds (a #Q s2) . n <= a #Q q
let n be Nat; :: thesis: (a #Q s2) . n <= a #Q q
s2 . n <= b by A9;
then s2 . n <= q by ;
then a #Q (s2 . n) <= a #Q q by ;
hence (a #Q s2) . n <= a #Q q by Def5; :: thesis: verum
end;
then A11: lim (a #Q s2) <= a #Q q by A1, A7, Th2, Th69;
a #Q s2 is convergent by A1, A7, Th69;
then a #R b <= a #Q q by A1, A7, A8, A11, Def6;
then A12: a #R b < a #Q p by ;
consider s1 being Rational_Sequence such that
A13: s1 is convergent and
A14: c = lim s1 and
A15: for n being Nat holds s1 . n >= c by Th68;
now :: thesis: for n being Nat holds (a #Q s1) . n >= a #Q p
let n be Nat; :: thesis: (a #Q s1) . n >= a #Q p
s1 . n >= c by A15;
then s1 . n >= p by ;
then a #Q (s1 . n) >= a #Q p by ;
hence (a #Q s1) . n >= a #Q p by Def5; :: thesis: verum
end;
then A16: lim (a #Q s1) >= a #Q p by A1, A13, Th1, Th69;
a #Q s1 is convergent by ;
then a #R c >= a #Q p by A1, A13, A14, A16, Def6;
hence a #R c > a #R b by ; :: thesis: verum