let a be ordinal number ; :: thesis: for f being Ordinal-Sequence st 1 in a & ( for b being ordinal number st b in dom f holds
f . b = exp a,b ) holds
f is increasing
let f be Ordinal-Sequence; :: thesis: ( 1 in a & ( for b being ordinal number st b in dom f holds
f . b = exp a,b ) implies f is increasing )
assume Z0: ( 1 in a & ( for b being ordinal number st b in dom f holds
f . b = exp a,b ) ) ; :: thesis: f is increasing
let b be ordinal number ; :: according to ORDINAL2:def 16 :: thesis: for b₁ being set holds
( not b in b₁ or not b₁ in proj1 f or f . b in f . b₁ )
let d be ordinal number ; :: thesis: ( not b in d or not d in proj1 f or f . b in f . d )
assume C1: ( b in d & d in dom f ) ; :: thesis: f . b in f . d
then ( f . b = exp a,b & f . d = exp a,d ) by Z0, ORDINAL1:19;
hence f . b in f . d by Z0, C1, ORDINAL4:24; :: thesis: verum