let X be set ; :: thesis: for A being Subset of X
for A1 being SetSequence of X st A1 is non-ascending holds
A (\) A1 is non-descending
let A be Subset of X; :: thesis: for A1 being SetSequence of X st A1 is non-ascending holds
A (\) A1 is non-descending
let A1 be SetSequence of X; :: thesis: ( A1 is non-ascending implies A (\) A1 is non-descending )
assume A1: A1 is non-ascending ; :: thesis: A (\) A1 is non-descending
for n, m being Nat st n <= m holds
(A (\) A1) . n c= (A (\) A1) . m

proof

let n, m be Nat; :: thesis: ( n <= m implies (A (\) A1) . n c= (A (\) A1) . m )
assume n <= m ; :: thesis: (A (\) A1) . n c= (A (\) A1) . m
then A1 . m c= A1 . n by A1, PROB_1:def 4;
then A \ (A1 . n) c= A \ (A1 . m) by XBOOLE_1:34;
then (A (\) A1) . n c= A \ (A1 . m) by Def7;
hence (A (\) A1) . n c= (A (\) A1) . m by Def7; :: thesis: verum

end;

hence A (\) A1 is non-descending by PROB_1:def 5; :: thesis: verum