let Y be set ; for h being PartFunc of REAL,REAL holds
( h | Y is non-increasing iff for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 <= r2 holds
h . r2 <= h . r1 )
let h be PartFunc of REAL,REAL; ( h | Y is non-increasing iff for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 <= r2 holds
h . r2 <= h . r1 )
thus
( h | Y is non-increasing implies for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 <= r2 holds
h . r2 <= h . r1 )
( ( for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 <= r2 holds
h . r2 <= h . r1 ) implies h | Y is non-increasing )
assume A4:
for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 <= r2 holds
h . r2 <= h . r1
; h | Y is non-increasing
let r1 be Real; RFUNCT_2:def 4 for r2 being Real st r1 in dom (h | Y) & r2 in dom (h | Y) & r1 < r2 holds
(h | Y) . r2 <= (h | Y) . r1
let r2 be Real; ( r1 in dom (h | Y) & r2 in dom (h | Y) & r1 < r2 implies (h | Y) . r2 <= (h | Y) . r1 )
assume that
A5:
( r1 in dom (h | Y) & r2 in dom (h | Y) )
and
A6:
r1 < r2
; (h | Y) . r2 <= (h | Y) . r1
A7:
( (h | Y) . r1 = h . r1 & (h | Y) . r2 = h . r2 )
by A5, FUNCT_1:47;
( r1 in Y /\ (dom h) & r2 in Y /\ (dom h) )
by A5, RELAT_1:61;
hence
(h | Y) . r2 <= (h | Y) . r1
by A4, A6, A7; verum