let Y be set ; :: thesis: for h being PartFunc of REAL,REAL holds
( h | Y is increasing iff for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 < r2 holds
h . r1 < h . r2 )

let h be PartFunc of REAL,REAL; :: thesis: ( h | Y is increasing iff for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 < r2 holds
h . r1 < h . r2 )

thus ( h | Y is 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 ) :: thesis: ( ( for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 < r2 holds
h . r1 < h . r2 ) implies h | Y is increasing )
proof
assume A1: h | Y is increasing ; :: thesis: for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 < r2 holds
h . r2 > h . r1

let r1, r2 be Real; :: thesis: ( r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 < r2 implies h . r2 > h . r1 )
assume that
A2: r1 in Y /\ (dom h) and
A3: r2 in Y /\ (dom h) and
A4: r1 < r2 ; :: thesis: h . r2 > h . r1
A5: r2 in dom (h | Y) by ;
then A6: (h | Y) . r2 = h . r2 by FUNCT_1:47;
A7: r1 in dom (h | Y) by ;
then (h | Y) . r1 = h . r1 by FUNCT_1:47;
hence h . r2 > h . r1 by A1, A4, A7, A5, A6; :: thesis: verum
end;
assume A8: for r1, r2 being Real st r1 in Y /\ (dom h) & r2 in Y /\ (dom h) & r1 < r2 holds
h . r2 > h . r1 ; :: thesis: h | Y is increasing
let r1 be Real; :: according to RFUNCT_2:def 1 :: thesis: for r2 being Real st r1 in dom (h | Y) & r2 in dom (h | Y) & r1 < r2 holds
(h | Y) . r1 < (h | Y) . r2

let r2 be Real; :: thesis: ( r1 in dom (h | Y) & r2 in dom (h | Y) & r1 < r2 implies (h | Y) . r1 < (h | Y) . r2 )
assume that
A9: ( r1 in dom (h | Y) & r2 in dom (h | Y) ) and
A10: r1 < r2 ; :: thesis: (h | Y) . r1 < (h | Y) . r2
A11: ( (h | Y) . r1 = h . r1 & (h | Y) . r2 = h . r2 ) by ;
( r1 in Y /\ (dom h) & r2 in Y /\ (dom h) ) by ;
hence (h | Y) . r1 < (h | Y) . r2 by A8, A10, A11; :: thesis: verum