let X be set ; :: thesis: for F being non empty ordered Subset-Family of X
for G being set st G in F holds
( G = max F iff for Y being set st Y in F holds
Y c= G )
let F be non empty ordered Subset-Family of X; :: thesis: for G being set st G in F holds
( G = max F iff for Y being set st Y in F holds
Y c= G )
let G be set ; :: thesis: ( G in F implies ( G = max F iff for Y being set st Y in F holds
Y c= G ) )
assume A1: G in F ; :: thesis: ( G = max F iff for Y being set st Y in F holds
Y c= G )
thus ( G = max F implies for Y being set st Y in F holds
Y c= G ) by ZFMISC_1:74; :: thesis: ( ( for Y being set st Y in F holds
Y c= G ) implies G = max F )
assume A2: for Y being set st Y in F holds
Y c= G ; :: thesis: G = max F
A3: G c= max F by A1, ZFMISC_1:74;
max F c= G by A2;
hence G = max F by A3, XBOOLE_0:def 10; :: thesis: verum