let s be FinSequence of A; :: thesis: ( s is descending implies s is weakly-descending )
assume s is descending ; :: thesis: s is weakly-descending
then A1: for n, m being Nat st n in dom s & m in dom s & n < m holds
s /. m <~ s /. n ;
for n, m being Nat st n in dom s & m in dom s & n < m holds
s /. m <= s /. n

proof

let n, m be Nat; :: thesis: ( n in dom s & m in dom s & n < m implies s /. m <= s /. n )
assume that
A2: ( n in dom s & m in dom s ) and
A3: n < m ; :: thesis: s /. m <= s /. n
s /. m <~ s /. n by A1, A2, A3;
hence s /. m <= s /. n ; :: thesis: verum

end;

hence s is weakly-descending ; :: thesis: verum