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

proof

let n, m be Nat; :: thesis: ( n <= m implies (A (\/) A1) . m c= (A (\/) A1) . n )
assume n <= m ; :: thesis: (A (\/) A1) . m c= (A (\/) A1) . n
then A1 . m c= A1 . n by A1, PROB_1:def 4;
then A \/ (A1 . m) c= A \/ (A1 . n) by XBOOLE_1:9;
then (A (\/) A1) . m c= A \/ (A1 . n) by Def6;
hence (A (\/) A1) . m c= (A (\/) A1) . n by Def6; :: thesis: verum

end;

hence A (\/) A1 is non-ascending by PROB_1:def 4; :: thesis: verum