thus ( s is non-decreasing 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-decreasing )

proof

assume A15: s is non-decreasing ; :: thesis: for n being Nat holds s . n <= s . (n + 1)
let n be Nat; :: thesis: s . n <= s . (n + 1)
A16: 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 A15, A16; :: thesis: verum

end;

assume A17: for n being Nat holds s . n <= s . (n + 1) ; :: thesis: s is non-decreasing
let e1 be ExtReal; :: according to VALUED_0:def 15 :: thesis: for b₁ being object holds
( not e1 in dom s or not b₁ in dom s or not e1 <= b₁ or s . e1 <= s . b₁ )
let e2 be ExtReal; :: thesis: ( not e1 in dom s or not e2 in dom s or not e1 <= e2 or s . e1 <= s . e2 )
assume ( e1 in dom s & e2 in dom s ) ; :: thesis: ( not e1 <= e2 or s . e1 <= s . e2 )
then reconsider m = e1, n = e2 as Nat ;
defpred S₁[ Nat] means ( m <= $1 implies s . m <= s . $1 );
A18: for j being Nat st S₁[j] holds
S₁[j + 1]

proof

let j be Nat; :: thesis: ( S₁[j] implies S₁[j + 1] )
assume A19: S₁[j] ; :: thesis: S₁[j + 1]
assume m <= j + 1 ; :: thesis: s . m <= s . (j + 1)
then A20: ( m < j + 1 or m = j + 1 ) by XXREAL_0:1;
s . j <= s . (j + 1) by A17;
hence s . m <= s . (j + 1) by A19, A20, NAT_1:13, XXREAL_0:2; :: thesis: verum

end;

assume A21: e1 <= e2 ; :: thesis: s . e1 <= s . e2
A22: S₁[ 0 ] ;
for j being Nat holds S₁[j] from NAT_1:sch 2(A22, A18);
then s . m <= s . n by A21;
hence s . e1 <= s . e2 ; :: thesis: verum