let A be closed-interval Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st f | A is non-decreasing & A c= dom f holds
rng (f | A) is bounded
let f be PartFunc of REAL ,REAL ; :: thesis: ( f | A is non-decreasing & A c= dom f implies rng (f | A) is bounded )
assume that
A1: f | A is non-decreasing and
A2: A c= dom f ; :: thesis: rng (f | A) is bounded
A3: dom (f | A) = (dom f) /\ A by RELAT_1:90
.= A by A2, XBOOLE_1:28 ;
f . (lower_bound A) is LowerBound of rng (f | A)

proof

lower_bound A <= upper_bound A by SEQ_4:24;
then lower_bound A in dom (f | A) by A3, INTEGRA2:1;
then A4: lower_bound A in (dom f) /\ A by RELAT_1:90;
let y be ext-real number ; :: according to XXREAL_2:def 2 :: thesis: ( not y in rng (f | A) or f . (lower_bound A) <= y )
assume y in rng (f | A) ; :: thesis: f . (lower_bound A) <= y
then consider x being Real such that
A5: x in dom (f | A) and
A6: y = (f | A) . x by PARTFUN1:26;
A7: x in (dom f) /\ A by A5, RELAT_1:90;
( y = f . x & x >= lower_bound A ) by A3, A5, A6, FUNCT_1:70, INTEGRA2:1;
hence f . (lower_bound A) <= y by A1, A7, A4, RFUNCT_2:48; :: thesis: verum

end;

then A8: rng (f | A) is bounded_below by XXREAL_2:def 9;
f . (upper_bound A) is UpperBound of rng (f | A)

proof

lower_bound A <= upper_bound A by SEQ_4:24;
then upper_bound A in dom (f | A) by A3, INTEGRA2:1;
then A9: upper_bound A in (dom f) /\ A by RELAT_1:90;
let y be ext-real number ; :: according to XXREAL_2:def 1 :: thesis: ( not y in rng (f | A) or y <= f . (upper_bound A) )
assume y in rng (f | A) ; :: thesis: y <= f . (upper_bound A)
then consider x being Real such that
A10: x in dom (f | A) and
A11: y = (f | A) . x by PARTFUN1:26;
A12: x in (dom f) /\ A by A10, RELAT_1:90;
( y = f . x & x <= upper_bound A ) by A3, A10, A11, FUNCT_1:70, INTEGRA2:1;
hence y <= f . (upper_bound A) by A1, A12, A9, RFUNCT_2:48; :: thesis: verum

end;

then rng (f | A) is bounded_above by XXREAL_2:def 10;
hence rng (f | A) is bounded by A8; :: thesis: verum