:: RFUNCT_4 semantic presentation
theorem Th1: :: RFUNCT_4:1
theorem Th2: :: RFUNCT_4:2
theorem Th3: :: RFUNCT_4:3
theorem Th4: :: RFUNCT_4:4
theorem Th5: :: RFUNCT_4:5
:: deftheorem Def1 defines is_strictly_convex_on RFUNCT_4:def 1 :
theorem Th6: :: RFUNCT_4:6
theorem Th7: :: RFUNCT_4:7
theorem Th8: :: RFUNCT_4:8
theorem Th9: :: RFUNCT_4:9
theorem Th10: :: RFUNCT_4:10
Lemma8:
for b1 being Real
for b2 being PartFunc of REAL , REAL
for b3 being set holds
( b2 is_strictly_convex_on b3 implies b2 - b1 is_strictly_convex_on b3 )
theorem Th11: :: RFUNCT_4:11
Lemma9:
for b1 being Real
for b2 being PartFunc of REAL , REAL
for b3 being set holds
( 0 < b1 & b2 is_strictly_convex_on b3 implies b1 (#) b2 is_strictly_convex_on b3 )
theorem Th12: :: RFUNCT_4:12
theorem Th13: :: RFUNCT_4:13
theorem Th14: :: RFUNCT_4:14
theorem Th15: :: RFUNCT_4:15
Lemma13:
for b1, b2, b3 being real number holds b3 * (b1 / b2) = (b3 * b1) / b2
by XCMPLX_1:75;
theorem Th16: :: RFUNCT_4:16
for b
1 being
PartFunc of
REAL ,
REAL for b
2 being
set holds
( b
1 is_convex_on b
2 iff ( b
2 c= dom b
1 & ( for b
3, b
4, b
5 being
Real holds
( b
3 in b
2 & b
4 in b
2 & b
5 in b
2 & b
3 < b
5 & b
5 < b
4 implies (
((b1 . b5) - (b1 . b3)) / (b5 - b3) <= ((b1 . b4) - (b1 . b3)) / (b4 - b3) &
((b1 . b4) - (b1 . b3)) / (b4 - b3) <= ((b1 . b4) - (b1 . b5)) / (b4 - b5) ) ) ) ) )
theorem Th17: :: RFUNCT_4:17
theorem Th18: :: RFUNCT_4:18
theorem Th19: :: RFUNCT_4:19
:: deftheorem Def2 defines is_quasiconvex_on RFUNCT_4:def 2 :
:: deftheorem Def3 defines is_strictly_quasiconvex_on RFUNCT_4:def 3 :
:: deftheorem Def4 defines is_strongly_quasiconvex_on RFUNCT_4:def 4 :
:: deftheorem Def5 defines is_upper_semicontinuous_in RFUNCT_4:def 5 :
:: deftheorem Def6 defines is_upper_semicontinuous_on RFUNCT_4:def 6 :
:: deftheorem Def7 defines is_lower_semicontinuous_in RFUNCT_4:def 7 :
:: deftheorem Def8 defines is_lower_semicontinuous_on RFUNCT_4:def 8 :
theorem Th20: :: RFUNCT_4:20
theorem Th21: :: RFUNCT_4:21
theorem Th22: :: RFUNCT_4:22
theorem Th23: :: RFUNCT_4:23
theorem Th24: :: RFUNCT_4:24
theorem Th25: :: RFUNCT_4:25
theorem Th26: :: RFUNCT_4:26