:: HAHNBAN semantic presentation

theorem Th1: :: HAHNBAN:1
canceled;

theorem Th2: :: HAHNBAN:2
canceled;

theorem Th3: :: HAHNBAN:3
for b1 being non empty set
for b2 being set holds
not ( b1 <> {b2} & ( for b3 being Element of b1 holds not b3 <> b2 ) )
proof end;

theorem Th4: :: HAHNBAN:4
for b1, b2 being set
for b3 being non empty Subset of (PFuncs b1,b2) holds
b3 is non empty functional set
proof end;

theorem Th5: :: HAHNBAN:5
for b1 being non empty functional set
for b2 being Function holds
( b2 = union b1 implies ( dom b2 = union { (dom b3) where B is Element of b1 : verum } & rng b2 = union { (rng b3) where B is Element of b1 : verum } ) )
proof end;

theorem Th6: :: HAHNBAN:6
for b1 being non empty Subset of ExtREAL holds
( ( for b2 being R_eal holds
( b2 in b1 implies b2 <= -infty ) ) implies b1 = {-infty } )
proof end;

theorem Th7: :: HAHNBAN:7
for b1 being non empty Subset of ExtREAL holds
( ( for b2 being R_eal holds
( b2 in b1 implies +infty <= b2 ) ) implies b1 = {+infty } )
proof end;

theorem Th8: :: HAHNBAN:8
for b1 being non empty Subset of ExtREAL
for b2 being R_eal holds
not ( b2 < sup b1 & ( for b3 being R_eal holds
not ( b3 in b1 & b2 < b3 ) ) )
proof end;

theorem Th9: :: HAHNBAN:9
for b1 being non empty Subset of ExtREAL
for b2 being R_eal holds
not ( inf b1 < b2 & ( for b3 being R_eal holds
not ( b3 in b1 & b3 < b2 ) ) )
proof end;

theorem Th10: :: HAHNBAN:10
for b1, b2 being non empty Subset of ExtREAL holds
( ( for b3, b4 being R_eal holds
( b3 in b1 & b4 in b2 implies b3 <= b4 ) ) implies sup b1 <= inf b2 )
proof end;

registration
let c1 be non empty set ;
cluster non empty c=-linear Element of bool a1;
existence
ex b1 being Subset of c1 st
( b1 is c=-linear & not b1 is empty )
proof end;
end;

theorem Th11: :: HAHNBAN:11
canceled;

theorem Th12: :: HAHNBAN:12
canceled;

theorem Th13: :: HAHNBAN:13
for b1, b2 being set
for b3 being c=-linear Subset of (PFuncs b1,b2) holds union b3 in PFuncs b1,b2
proof end;

theorem Th14: :: HAHNBAN:14
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds the carrier of b2 c= the carrier of (b2 + b3)
proof end;

theorem Th15: :: HAHNBAN:15
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds
( b1 is_the_direct_sum_of b2,b3 implies for b4, b5, b6 being VECTOR of b1 holds
( b5 in b2 & b6 in b3 & b4 = b5 + b6 implies b4 |-- b2,b3 = [b5,b6] ) )
proof end;

theorem Th16: :: HAHNBAN:16
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds
( b1 is_the_direct_sum_of b2,b3 implies for b4, b5, b6 being VECTOR of b1 holds
( b4 |-- b2,b3 = [b5,b6] implies b4 = b5 + b6 ) )
proof end;

theorem Th17: :: HAHNBAN:17
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds
( b1 is_the_direct_sum_of b2,b3 implies for b4, b5, b6 being VECTOR of b1 holds
( b4 |-- b2,b3 = [b5,b6] implies ( b5 in b2 & b6 in b3 ) ) )
proof end;

theorem Th18: :: HAHNBAN:18
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds
( b1 is_the_direct_sum_of b2,b3 implies for b4, b5, b6 being VECTOR of b1 holds
( b4 |-- b2,b3 = [b5,b6] implies b4 |-- b3,b2 = [b6,b5] ) )
proof end;

theorem Th19: :: HAHNBAN:19
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds
( b1 is_the_direct_sum_of b2,b3 implies for b4 being VECTOR of b1 holds
( b4 in b2 implies b4 |-- b2,b3 = [b4,(0. b1)] ) )
proof end;

theorem Th20: :: HAHNBAN:20
for b1 being RealLinearSpace
for b2, b3 being Subspace of b1 holds
( b1 is_the_direct_sum_of b2,b3 implies for b4 being VECTOR of b1 holds
( b4 in b3 implies b4 |-- b2,b3 = [(0. b1),b4] ) )
proof end;

theorem Th21: :: HAHNBAN:21
for b1 being RealLinearSpace
for b2 being Subspace of b1
for b3 being Subspace of b2
for b4 being VECTOR of b1 holds
( b4 in b3 implies b4 is VECTOR of b2 )
proof end;

theorem Th22: :: HAHNBAN:22
for b1 being RealLinearSpace
for b2, b3, b4 being Subspace of b1
for b5, b6 being Subspace of b4 holds
( b5 = b2 & b6 = b3 implies b5 + b6 = b2 + b3 )
proof end;

theorem Th23: :: HAHNBAN:23
for b1 being RealLinearSpace
for b2 being Subspace of b1
for b3 being VECTOR of b1
for b4 being VECTOR of b2 holds
( b3 = b4 implies Lin {b4} = Lin {b3} )
proof end;

theorem Th24: :: HAHNBAN:24
for b1 being RealLinearSpace
for b2 being VECTOR of b1
for b3 being Subspace of b1 holds
( not b2 in b3 implies for b4 being VECTOR of (b3 + (Lin {b2}))
for b5 being Subspace of b3 + (Lin {b2}) holds
( b2 = b4 & b5 = b3 implies b3 + (Lin {b2}) is_the_direct_sum_of b5, Lin {b4} ) )
proof end;

theorem Th25: :: HAHNBAN:25
for b1 being RealLinearSpace
for b2 being VECTOR of b1
for b3 being Subspace of b1
for b4 being VECTOR of (b3 + (Lin {b2}))
for b5 being Subspace of b3 + (Lin {b2}) holds
( b2 = b4 & b3 = b5 & not b2 in b3 implies b4 |-- b5,(Lin {b4}) = [(0. b5),b4] )
proof end;

theorem Th26: :: HAHNBAN:26
for b1 being RealLinearSpace
for b2 being VECTOR of b1
for b3 being Subspace of b1
for b4 being VECTOR of (b3 + (Lin {b2}))
for b5 being Subspace of b3 + (Lin {b2}) holds
( b2 = b4 & b3 = b5 & not b2 in b3 implies for b6 being VECTOR of (b3 + (Lin {b2})) holds
( b6 in b3 implies b6 |-- b5,(Lin {b4}) = [b6,(0. b1)] ) )
proof end;

theorem Th27: :: HAHNBAN:27
for b1 being RealLinearSpace
for b2 being VECTOR of b1
for b3, b4 being Subspace of b1 holds
ex b5, b6 being VECTOR of b1 st b2 |-- b3,b4 = [b5,b6]
proof end;

theorem Th28: :: HAHNBAN:28
for b1 being RealLinearSpace
for b2 being VECTOR of b1
for b3 being Subspace of b1
for b4 being VECTOR of (b3 + (Lin {b2}))
for b5 being Subspace of b3 + (Lin {b2}) holds
( b2 = b4 & b3 = b5 & not b2 in b3 implies for b6 being VECTOR of (b3 + (Lin {b2})) holds
ex b7 being VECTOR of b3ex b8 being Real st b6 |-- b5,(Lin {b4}) = [b7,(b8 * b2)] )
proof end;

theorem Th29: :: HAHNBAN:29
for b1 being RealLinearSpace
for b2 being VECTOR of b1
for b3 being Subspace of b1
for b4 being VECTOR of (b3 + (Lin {b2}))
for b5 being Subspace of b3 + (Lin {b2}) holds
( b2 = b4 & b3 = b5 & not b2 in b3 implies for b6, b7 being VECTOR of (b3 + (Lin {b2}))
for b8, b9 being VECTOR of b3
for b10, b11 being Real holds
( b6 |-- b5,(Lin {b4}) = [b8,(b10 * b2)] & b7 |-- b5,(Lin {b4}) = [b9,(b11 * b2)] implies (b6 + b7) |-- b5,(Lin {b4}) = [(b8 + b9),((b10 + b11) * b2)] ) )
proof end;

theorem Th30: :: HAHNBAN:30
for b1 being RealLinearSpace
for b2 being VECTOR of b1
for b3 being Subspace of b1
for b4 being VECTOR of (b3 + (Lin {b2}))
for b5 being Subspace of b3 + (Lin {b2}) holds
( b2 = b4 & b3 = b5 & not b2 in b3 implies for b6 being VECTOR of (b3 + (Lin {b2}))
for b7 being VECTOR of b3
for b8, b9 being Real holds
( b6 |-- b5,(Lin {b4}) = [b7,(b9 * b2)] implies (b8 * b6) |-- b5,(Lin {b4}) = [(b8 * b7),((b8 * b9) * b2)] ) )
proof end;

definition
let c1 be RLSStruct ;
mode Functional is Function of the carrier of a1, REAL ;
end;

definition
let c1 be RealLinearSpace;
let c2 be Functional of c1;
canceled;
canceled;
attr a2 is subadditive means :Def3: :: HAHNBAN:def 3
for b1, b2 being VECTOR of a1 holds a2 . (b1 + b2) <= (a2 . b1) + (a2 . b2);
attr a2 is additive means :Def4: :: HAHNBAN:def 4
for b1, b2 being VECTOR of a1 holds a2 . (b1 + b2) = (a2 . b1) + (a2 . b2);
attr a2 is homogeneous means :Def5: :: HAHNBAN:def 5
for b1 being VECTOR of a1
for b2 being Real holds a2 . (b2 * b1) = b2 * (a2 . b1);
attr a2 is positively_homogeneous means :Def6: :: HAHNBAN:def 6
for b1 being VECTOR of a1
for b2 being Real holds
( b2 > 0 implies a2 . (b2 * b1) = b2 * (a2 . b1) );
attr a2 is semi-homogeneous means :Def7: :: HAHNBAN:def 7
for b1 being VECTOR of a1
for b2 being Real holds
( b2 >= 0 implies a2 . (b2 * b1) = b2 * (a2 . b1) );
attr a2 is absolutely_homogeneous means :Def8: :: HAHNBAN:def 8
for b1 being VECTOR of a1
for b2 being Real holds a2 . (b2 * b1) = (abs b2) * (a2 . b1);
attr a2 is 0-preserving means :Def9: :: HAHNBAN:def 9
a2 . (0. a1) = 0;
end;

:: deftheorem Def1 HAHNBAN:def 1 :
canceled;

:: deftheorem Def2 HAHNBAN:def 2 :
canceled;

:: deftheorem Def3 defines subadditive HAHNBAN:def 3 :
for b1 being RealLinearSpace
for b2 being Functional of b1 holds
( b2 is subadditive iff for b3, b4 being VECTOR of b1 holds b2 . (b3 + b4) <= (b2 . b3) + (b2 . b4) );

:: deftheorem Def4 defines additive HAHNBAN:def 4 :
for b1 being RealLinearSpace
for b2 being Functional of b1 holds
( b2 is additive iff for b3, b4 being VECTOR of b1 holds b2 . (b3 + b4) = (b2 . b3) + (b2 . b4) );

:: deftheorem Def5 defines homogeneous HAHNBAN:def 5 :
for b1 being RealLinearSpace
for b2 being Functional of b1 holds
( b2 is homogeneous iff for b3 being VECTOR of b1
for b4 being Real holds b2 . (b4 * b3) = b4 * (b2 . b3) );

:: deftheorem Def6 defines positively_homogeneous HAHNBAN:def 6 :
for b1 being RealLinearSpace
for b2 being Functional of b1 holds
( b2 is positively_homogeneous iff for b3 being VECTOR of b1
for b4 being Real holds
( b4 > 0 implies b2 . (b4 * b3) = b4 * (b2 . b3) ) );

:: deftheorem Def7 defines semi-homogeneous HAHNBAN:def 7 :
for b1 being RealLinearSpace
for b2 being Functional of b1 holds
( b2 is semi-homogeneous iff for b3 being VECTOR of b1
for b4 being Real holds
( b4 >= 0 implies b2 . (b4 * b3) = b4 * (b2 . b3) ) );

:: deftheorem Def8 defines absolutely_homogeneous HAHNBAN:def 8 :
for b1 being RealLinearSpace
for b2 being Functional of b1 holds
( b2 is absolutely_homogeneous iff for b3 being VECTOR of b1
for b4 being Real holds b2 . (b4 * b3) = (abs b4) * (b2 . b3) );

:: deftheorem Def9 defines 0-preserving HAHNBAN:def 9 :
for b1 being RealLinearSpace
for b2 being Functional of b1 holds
( b2 is 0-preserving iff b2 . (0. b1) = 0 );

registration
let c1 be RealLinearSpace;
cluster additive -> subadditive M5(the carrier of a1, REAL );
coherence
for b1 being Functional of c1 holds
( b1 is additive implies b1 is subadditive )
proof end;
cluster homogeneous -> positively_homogeneous M5(the carrier of a1, REAL );
coherence
for b1 being Functional of c1 holds
( b1 is homogeneous implies b1 is positively_homogeneous )
proof end;
cluster semi-homogeneous -> positively_homogeneous M5(the carrier of a1, REAL );
coherence
for b1 being Functional of c1 holds
( b1 is semi-homogeneous implies b1 is positively_homogeneous )
proof end;
cluster semi-homogeneous -> 0-preserving M5(the carrier of a1, REAL );
coherence
for b1 being Functional of c1 holds
( b1 is semi-homogeneous implies b1 is 0-preserving )
proof end;
cluster absolutely_homogeneous -> semi-homogeneous M5(the carrier of a1, REAL );
coherence
for b1 being Functional of c1 holds
( b1 is absolutely_homogeneous implies b1 is semi-homogeneous )
proof end;
cluster positively_homogeneous 0-preserving -> semi-homogeneous M5(the carrier of a1, REAL );
coherence
for b1 being Functional of c1 holds
( b1 is 0-preserving & b1 is positively_homogeneous implies b1 is semi-homogeneous )
proof end;
end;

registration
let c1 be RealLinearSpace;
cluster subadditive additive homogeneous positively_homogeneous semi-homogeneous absolutely_homogeneous 0-preserving M5(the carrier of a1, REAL );
existence
ex b1 being Functional of c1 st
( b1 is additive & b1 is absolutely_homogeneous & b1 is homogeneous )
proof end;
end;

definition
let c1 be RealLinearSpace;
mode Banach-Functional is subadditive positively_homogeneous Functional of a1;
mode linear-Functional is additive homogeneous Functional of a1;
end;

theorem Th31: :: HAHNBAN:31
for b1 being RealLinearSpace
for b2 being homogeneous Functional of b1
for b3 being VECTOR of b1 holds b2 . (- b3) = - (b2 . b3)
proof end;

theorem Th32: :: HAHNBAN:32
for b1 being RealLinearSpace
for b2 being linear-Functional of b1
for b3, b4 being VECTOR of b1 holds b2 . (b3 - b4) = (b2 . b3) - (b2 . b4)
proof end;

theorem Th33: :: HAHNBAN:33
for b1 being RealLinearSpace
for b2 being additive Functional of b1 holds b2 . (0. b1) = 0
proof end;

theorem Th34: :: HAHNBAN:34
for b1 being RealLinearSpace
for b2 being Subspace of b1
for b3 being linear-Functional of b2
for b4 being VECTOR of b1
for b5 being VECTOR of (b2 + (Lin {b4})) holds
( b4 = b5 & not b4 in b2 implies for b6 being Real holds
ex b7 being linear-Functional of (b2 + (Lin {b4})) st
( b7 | the carrier of b2 = b3 & b7 . b5 = b6 ) )
proof end;

Lemma36: for b1 being RealLinearSpace
for b2 being Subspace of b1
for b3 being VECTOR of b1 holds
( not b3 in the carrier of b2 implies for b4 being Banach-Functional of b1
for b5 being linear-Functional of b2 holds
not ( ( for b6 being VECTOR of b2
for b7 being VECTOR of b1 holds
( b6 = b7 implies b5 . b6 <= b4 . b7 ) ) & ( for b6 being linear-Functional of (b2 + (Lin {b3})) holds
not ( b6 | the carrier of b2 = b5 & ( for b7 being VECTOR of (b2 + (Lin {b3}))
for b8 being VECTOR of b1 holds
( b7 = b8 implies b6 . b7 <= b4 . b8 ) ) ) ) ) )
proof end;

Lemma37: for b1 being RealLinearSpace holds
RLSStruct(# the carrier of b1,the Zero of b1,the add of b1,the Mult of b1 #) is RealLinearSpace
proof end;

Lemma38: for b1, b2, b3 being RealLinearSpace holds
( b2 = RLSStruct(# the carrier of b1,the Zero of b1,the add of b1,the Mult of b1 #) implies for b4 being Subspace of b1 holds
( b3 = RLSStruct(# the carrier of b4,the Zero of b4,the add of b4,the Mult of b4 #) implies b3 is Subspace of b2 ) )
proof end;

Lemma39: for b1, b2 being RealLinearSpace holds
( b2 = RLSStruct(# the carrier of b1,the Zero of b1,the add of b1,the Mult of b1 #) implies for b3 being linear-Functional of b2 holds
b3 is linear-Functional of b1 )
proof end;

Lemma40: for b1, b2 being RealLinearSpace holds
( b2 = RLSStruct(# the carrier of b1,the Zero of b1,the add of b1,the Mult of b1 #) implies for b3 being linear-Functional of b1 holds
b3 is linear-Functional of b2 )
proof end;

theorem Th35: :: HAHNBAN:35
for b1 being RealLinearSpace
for b2 being Subspace of b1
for b3 being Banach-Functional of b1
for b4 being linear-Functional of b2 holds
not ( ( for b5 being VECTOR of b2
for b6 being VECTOR of b1 holds
( b5 = b6 implies b4 . b5 <= b3 . b6 ) ) & ( for b5 being linear-Functional of b1 holds
not ( b5 | the carrier of b2 = b4 & ( for b6 being VECTOR of b1 holds b5 . b6 <= b3 . b6 ) ) ) )
proof end;

theorem Th36: :: HAHNBAN:36
for b1 being RealNormSpace holds
the norm of b1 is subadditive absolutely_homogeneous Functional of b1
proof end;

theorem Th37: :: HAHNBAN:37
for b1 being RealNormSpace
for b2 being Subspace of b1
for b3 being linear-Functional of b2 holds
not ( ( for b4 being VECTOR of b2
for b5 being VECTOR of b1 holds
( b4 = b5 implies b3 . b4 <= ||.b5.|| ) ) & ( for b4 being linear-Functional of b1 holds
not ( b4 | the carrier of b2 = b3 & ( for b5 being VECTOR of b1 holds b4 . b5 <= ||.b5.|| ) ) ) )
proof end;