let a be ordinal number ; :: thesis: for f being Ordinal-Sequence st 0 in a & ( for b being ordinal number st b in dom f holds
f . b = exp a,b ) holds
f is non-decreasing
let f be Ordinal-Sequence; :: thesis: ( 0 in a & ( for b being ordinal number st b in dom f holds
f . b = exp a,b ) implies f is non-decreasing )
assume Z0: ( 0 in a & ( for b being ordinal number st b in dom f holds
f . b = exp a,b ) ) ; :: thesis: f is non-decreasing
let b be ordinal number ; :: according to ORDINAL5:def 2 :: thesis: for b being ordinal number st b in b & b in dom f holds
f . b c= f . b
let d be ordinal number ; :: thesis: ( b in d & d in dom f implies f . b c= f . d )
assume C1: ( b in d & d in dom f ) ; :: thesis: f . b c= f . d
then b in dom f by ORDINAL1:19;
then ( b c= d & f . b = exp a,b & f . d = exp a,d ) by Z0, C1, ORDINAL1:def 2;
hence f . b c= f . d by Z0, ORDINAL4:27; :: thesis: verum