let f be Real_Sequence; :: thesis: ( f is non-decreasing iff for n being Element of NAT holds f . n <= f . (n + 1) )
thus ( f is non-decreasing implies for n being Element of NAT holds f . n <= f . (n + 1) ) :: thesis: ( ( for n being Element of NAT holds f . n <= f . (n + 1) ) implies f is non-decreasing )

proof

assume A1: f is non-decreasing ; :: thesis: for n being Element of NAT holds f . n <= f . (n + 1)
let n be Element of NAT ; :: thesis: f . n <= f . (n + 1)
( dom f = NAT & n < n + 1 ) by FUNCT_2:def 1, NAT_1:13;
hence f . n <= f . (n + 1) by A1, Def3; :: thesis: verum

end;

assume A2: for n being Element of NAT holds f . n <= f . (n + 1) ; :: thesis: f is non-decreasing
let m be Element of NAT ; :: according to SEQM_3:def 3 :: thesis: for n being Element of NAT st m in dom f & n in dom f & m <= n holds
f . m <= f . n
let n be Element of NAT ; :: thesis: ( m in dom f & n in dom f & m <= n implies f . m <= f . n )
assume that
m in dom f and
n in dom f and
A3: m <= n ; :: thesis: f . m <= f . n
consider k being Nat such that
A4: n = m + k by A3, NAT_1:10;
k in NAT by ORDINAL1:def 12;
hence f . m <= f . n by A2, A4, Lm4; :: thesis: verum