let s be FinSequence of A; :: thesis: ( s is one-to-one & s is weakly-descending implies s is descending )
assume that
A6: s is one-to-one and
A7: s is weakly-descending ; :: thesis: s is descending
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
A8: ( n in dom s & m in dom s ) and
A9: n < m ; :: thesis: s /. m <~ s /. n
A10: s /. m <= s /. n by A8, A9, A7;
not s /. n <= s /. m

proof

assume s /. n <= s /. m ; :: thesis: contradiction
then s /. m = s /. n by A10, ORDERS_2:2;
then s . m = s /. n by A8, PARTFUN1:def 6;
then s . m = s . n by A8, PARTFUN1:def 6;
hence contradiction by A8, A9, A6, FUNCT_1:def 4; :: thesis: verum

end;

hence s /. m <~ s /. n by A10; :: thesis: verum

end;

hence s is descending ; :: thesis: verum