thus ( s is non-increasing implies for n being Nat holds s . n >= s . (n + 1) ) :: thesis: ( ( for n being Nat holds s . n >= s . (n + 1) ) implies s is non-increasing )

proof

assume A23: s is non-increasing ; :: thesis: for n being Nat holds s . n >= s . (n + 1)
let n be Nat; :: thesis: s . n >= s . (n + 1)
A24: n < n + 1 by NAT_1:13;
( dom s = NAT & n in NAT ) by FUNCT_2:def 1, ORDINAL1:def 12;
hence s . n >= s . (n + 1) by A23, A24; :: thesis: verum

end;

assume A25: for n being Nat holds s . n >= s . (n + 1) ; :: thesis: s is non-increasing
let e1 be ExtReal; :: according to VALUED_0:def 16 :: thesis: for b₁ being object holds
( not e1 in dom s or not b₁ in dom s or not e1 <= b₁ or s . b₁ <= s . e1 )
let e2 be ExtReal; :: thesis: ( not e1 in dom s or not e2 in dom s or not e1 <= e2 or s . e2 <= s . e1 )
assume ( e1 in dom s & e2 in dom s ) ; :: thesis: ( not e1 <= e2 or s . e2 <= s . e1 )
then reconsider m = e1, n = e2 as Nat ;
defpred S₁[ Nat] means ( m <= $1 implies s . m >= s . $1 );
A26: for j being Nat st S₁[j] holds
S₁[j + 1]

proof

let j be Nat; :: thesis: ( S₁[j] implies S₁[j + 1] )
assume A27: S₁[j] ; :: thesis: S₁[j + 1]
assume m <= j + 1 ; :: thesis: s . m >= s . (j + 1)
then A28: ( m < j + 1 or m = j + 1 ) by XXREAL_0:1;
s . j >= s . (j + 1) by A25;
hence s . m >= s . (j + 1) by A27, A28, NAT_1:13, XXREAL_0:2; :: thesis: verum

end;

assume A29: e1 <= e2 ; :: thesis: s . e2 <= s . e1
A30: S₁[ 0 ] ;
for j being Nat holds S₁[j] from NAT_1:sch 2(A30, A26);
then s . m >= s . n by A29;
hence s . e2 <= s . e1 ; :: thesis: verum