thus ( s is 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 decreasing )

proof

assume A8: s is decreasing ; :: thesis: for n being Nat holds s . n > s . (n + 1)
let n be Nat; :: thesis: s . n > s . (n + 1)
A9: 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 A8, A9; :: thesis: verum

end;

assume A10: for n being Nat holds s . n > s . (n + 1) ; :: thesis: s is decreasing
let e1 be ExtReal; :: according to VALUED_0:def 14 :: thesis: for b₁ being object holds
( not e1 in dom s or not b₁ in dom s or b₁ <= e1 or not s . e1 <= s . b₁ )
let e2 be ExtReal; :: thesis: ( not e1 in dom s or not e2 in dom s or e2 <= e1 or not s . e1 <= s . e2 )
assume ( e1 in dom s & e2 in dom s ) ; :: thesis: ( e2 <= e1 or not s . e1 <= s . e2 )
then reconsider m = e1, n = e2 as Nat ;
defpred S₁[ Nat] means ( m < $1 implies s . m > s . $1 );
A11: for j being Nat st S₁[j] holds
S₁[j + 1]

proof

let j be Nat; :: thesis: ( S₁[j] implies S₁[j + 1] )
assume A12: S₁[j] ; :: thesis: S₁[j + 1]
assume m < j + 1 ; :: thesis: s . m > s . (j + 1)
then m <= j by NAT_1:13;
then ( s . m > s . j or m = j ) by A12, XXREAL_0:1;
hence s . m > s . (j + 1) by A10, XXREAL_0:2; :: thesis: verum

end;

assume A13: e1 < e2 ; :: thesis: not s . e1 <= s . e2
A14: S₁[ 0 ] ;
for j being Nat holds S₁[j] from NAT_1:sch 2(A14, A11);
then s . m > s . n by A13;
hence not s . e1 <= s . e2 ; :: thesis: verum