let n be Element of NAT ; :: thesis: for seq being Real_Sequence st seq is non-increasing holds
(superior_realsequence seq) . (n + 1) <= seq . n
let seq be Real_Sequence; :: thesis: ( seq is non-increasing implies (superior_realsequence seq) . (n + 1) <= seq . n )
assume A1: seq is non-increasing ; :: thesis: (superior_realsequence seq) . (n + 1) <= seq . n
reconsider Y1 = { (seq . k) where k is Element of NAT : n + 1 <= k } as Subset of REAL by Th29;
A2: (superior_realsequence seq) . (n + 1) = sup Y1 by Def5;
sup Y1 = seq . (n + 1) by A1, Th35;
hence (superior_realsequence seq) . (n + 1) <= seq . n by A1, A2, SEQM_3:def 14; :: thesis: verum