:: CONVEX2 semantic presentation
theorem Th1: :: CONVEX2:1
theorem Th2: :: CONVEX2:2
theorem Th3: :: CONVEX2:3
theorem Th4: :: CONVEX2:4
theorem Th5: :: CONVEX2:5
Lemma51:
for V being RealLinearSpace
for M being convex Subset of V
for N being Subset of V
for L being Linear_Combination of N st L is convex & N c= M holds
Sum L in M
Lemma136:
for V being RealLinearSpace
for M being Subset of V st ( for N being Subset of V
for L being Linear_Combination of N st L is convex & N c= M holds
Sum L in M ) holds
M is convex
theorem Th6: :: CONVEX2:6
:: deftheorem Def1 defines LinComb CONVEX2:def 1 :
Lemma141:
for V being RealLinearSpace ex L being Linear_Combination of V st L is convex
Lemma142:
for V being RealLinearSpace
for M being non empty Subset of V ex L being Linear_Combination of M st L is convex
Lemma143:
for V being RealLinearSpace
for W1, W2 being Subspace of V holds Up (W1 + W2) = (Up W1) + (Up W2)
Lemma146:
for V being RealLinearSpace
for W1, W2 being Subspace of V holds Up (W1 /\ W2) = (Up W1) /\ (Up W2)
theorem Th7: :: CONVEX2:7
theorem Th8: :: CONVEX2:8
Lemma149:
for V being RealLinearSpace
for L1, L2 being Convex_Combination of V
for a, b being Real st a * b > 0 holds
Carrier ((a * L1) + (b * L2)) = (Carrier (a * L1)) \/ (Carrier (b * L2))
Lemma153:
for F, G being Function st F,G are_fiberwise_equipotent holds
for x1, x2 being set st x1 in dom F & x2 in dom F & x1 <> x2 holds
ex z1, z2 being set st
( z1 in dom G & z2 in dom G & z1 <> z2 & F . x1 = G . z1 & F . x2 = G . z2 )
Lemma160:
for V being RealLinearSpace
for L being Linear_Combination of V
for A being Subset of V st A c= Carrier L holds
ex L1 being Linear_Combination of V st Carrier L1 = A
theorem Th9: :: CONVEX2:9
theorem Th10: :: CONVEX2:10
theorem Th11: :: CONVEX2:11
theorem Th12: :: CONVEX2:12
theorem Th13: :: CONVEX2:13
theorem Th14: :: CONVEX2:14
theorem Th15: :: CONVEX2:15
theorem Th16: :: CONVEX2:16
theorem Th17: :: CONVEX2:17