:: VECTSP_7 semantic presentation

begin

definition
let GF be ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ;
let V be ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ;
let IT be ( ( ) ( ) Subset of ) ;
attr IT is linearly-independent means :: VECTSP_7:def 1
 for l being ( ( ) ( Relation-like the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) -defined the carrier of GF : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Linear_Combination of IT : ( ( Function-like quasi_total ) ( Relation-like [:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) -defined V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) -valued Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) st Sum l : ( ( ) ( Relation-like the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like quasi_total ) Linear_Combination of IT : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) = 0. V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( V47(V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) holds
Carrier l : ( ( ) ( Relation-like the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like quasi_total ) Linear_Combination of IT : ( ( ) ( ) Subset of ) ) : ( ( finite ) ( finite ) Element of bool the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = {} : ( ( ) ( empty finite V35() ) set ) ;
end;

notation
let GF be ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ;
let V be ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ;
let IT be ( ( ) ( ) Subset of ) ;
antonym linearly-dependent IT for linearly-independent ;
end;

theorem :: VECTSP_7:1
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A, B being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) c= B : ( ( ) ( ) Subset of ) & B : ( ( ) ( ) Subset of ) is linearly-independent holds
A : ( ( ) ( ) Subset of ) is linearly-independent ;

theorem :: VECTSP_7:2
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) is linearly-independent holds
not 0. V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( V47(b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) in A : ( ( ) ( ) Subset of ) ;

registration
let GF be ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ;
let V be ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ;
cluster empty -> linearly-independent for ( ( ) ( ) Element of bool the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let GF be ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ;
let V be ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ;
cluster finite linearly-independent for ( ( ) ( ) Element of bool the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: VECTSP_7:3
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds
( {v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty finite ) Element of bool the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is linearly-independent iff v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) <> 0. V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( V47(b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP_7:4
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for v1, v2 being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) st {v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty finite ) Element of bool the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is linearly-independent holds
v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) <> 0. V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( V47(b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_7:5
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for v1, v2 being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds
( v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) <> v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) & {v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty finite ) Element of bool the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is linearly-independent iff ( v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) <> 0. V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( V47(b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) & ( for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) holds v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) <> a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: VECTSP_7:6
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for v1, v2 being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds
( ( v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) <> v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) & {v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty finite ) Element of bool the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is linearly-independent ) iff for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( V47(b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) = 0. GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( V47(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( non empty non trivial ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) = 0. GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( V47(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

definition
let GF be ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ;
let V be ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ;
let A be ( ( ) ( ) Subset of ) ;
func Lin A -> ( ( strict ) ( non empty V68() strict V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) Subspace of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) ) means :: VECTSP_7:def 2
 the carrier of it : ( ( ) ( ) Element of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) ) : ( ( ) ( ) set ) = { (Sum l : ( ( ) ( Relation-like the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like quasi_total ) Linear_Combination of A : ( ( ) ( ) Subset of ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) ) where l is ( ( ) ( Relation-like the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) -defined the carrier of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) : ( ( ) ( non empty non trivial ) set ) -valued Function-like quasi_total ) Linear_Combination of A : ( ( Function-like quasi_total ) ( Relation-like [:V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) ,V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) :] : ( ( ) ( ) set ) -defined V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) -valued Function-like quasi_total ) Element of bool [:[:V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) ,V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) :] : ( ( ) ( ) set ) ,V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : verum } ;
end;

theorem :: VECTSP_7:7
for x being ( ( ) ( ) set )
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of ) holds
( x : ( ( ) ( ) set ) in Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b3 : ( ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) iff ex l being ( ( ) ( Relation-like the carrier of b3 : ( ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like quasi_total ) Linear_Combination of A : ( ( ) ( ) Subset of ) ) st x : ( ( ) ( ) set ) = Sum l : ( ( ) ( Relation-like the carrier of b3 : ( ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like quasi_total ) Linear_Combination of b4 : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP_7:8
for x being ( ( ) ( ) set )
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of ) st x : ( ( ) ( ) set ) in A : ( ( ) ( ) Subset of ) holds
x : ( ( ) ( ) set ) in Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b3 : ( ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b2 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ;

theorem :: VECTSP_7:9
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) holds Lin ({} the carrier of V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( empty proper finite V35() linearly-independent ) Element of bool the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = (0). V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ;

theorem :: VECTSP_7:10
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of ) holds
( not Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = (0). V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) or A : ( ( ) ( ) Subset of ) = {} : ( ( ) ( empty finite V35() ) set ) or A : ( ( ) ( ) Subset of ) = {(0. V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) : ( ( ) ( V47(b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty finite ) Element of bool the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP_7:11
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of )
for W being ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) st A : ( ( ) ( ) Subset of ) = the carrier of W : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) : ( ( ) ( non empty ) set ) holds
Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = W : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ;

theorem :: VECTSP_7:12
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) = the carrier of V : ( ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) holds
Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = V : ( ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ;

theorem :: VECTSP_7:13
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A, B being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) c= B : ( ( ) ( ) Subset of ) holds
Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) is ( ( ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of Lin B : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ) ;

theorem :: VECTSP_7:14
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A, B being ( ( ) ( ) Subset of ) st Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = V : ( ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) & A : ( ( ) ( ) Subset of ) c= B : ( ( ) ( ) Subset of ) holds
Lin B : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = V : ( ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ;

theorem :: VECTSP_7:15
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A, B being ( ( ) ( ) Subset of ) holds Lin (A : ( ( ) ( ) Subset of ) \/ B : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = (Lin A : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) + (Lin B : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ;

theorem :: VECTSP_7:16
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A, B being ( ( ) ( ) Subset of ) holds Lin (A : ( ( ) ( ) Subset of ) /\ B : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) is ( ( ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of (Lin A : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) /\ (Lin B : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ) ;

theorem :: VECTSP_7:17
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) is linearly-independent holds
ex B being ( ( ) ( ) Subset of ) st
( A : ( ( ) ( ) Subset of ) c= B : ( ( ) ( ) Subset of ) & B : ( ( ) ( ) Subset of ) is linearly-independent & Lin B : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = VectSpStr(# the carrier of V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) , the U5 of V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) ) , the lmult of V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) VectSpStr over b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ;

theorem :: VECTSP_7:18
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of ) st Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) holds
ex B being ( ( ) ( ) Subset of ) st
( B : ( ( ) ( ) Subset of ) c= A : ( ( ) ( ) Subset of ) & B : ( ( ) ( ) Subset of ) is linearly-independent & Lin B : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ;

definition
let GF be ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ;
let V be ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ;
mode Basis of V -> ( ( ) ( ) Subset of ) means :: VECTSP_7:def 3
( it : ( ( Function-like quasi_total ) ( Relation-like [:V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) ,V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) :] : ( ( ) ( ) set ) -defined V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) -valued Function-like quasi_total ) Element of bool [:[:V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) ,V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) :] : ( ( ) ( ) set ) ,V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) is linearly-independent & Lin it : ( ( Function-like quasi_total ) ( Relation-like [:V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) ,V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) :] : ( ( ) ( ) set ) -defined V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) -valued Function-like quasi_total ) Element of bool [:[:V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) ,V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) :] : ( ( ) ( ) set ) ,V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty V68() strict V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) Subspace of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) ) = VectSpStr(# the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) , the U5 of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) ) , the lmult of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) : ( ( ) ( non empty non trivial ) set ) , the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) : ( ( ) ( non empty non trivial ) set ) , the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) ( non empty V68() V105(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V106(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V107(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V108(GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) V115() V116() V117() ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) VectSpStr over GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) L11()) ) );
end;

theorem :: VECTSP_7:19
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) is linearly-independent holds
ex I being ( ( ) ( ) Basis of V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) st A : ( ( ) ( ) Subset of ) c= I : ( ( ) ( ) Basis of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) ;

theorem :: VECTSP_7:20
for GF being ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field)
for V being ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of GF : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) )
for A being ( ( ) ( ) Subset of ) st Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty V68() strict V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) Subspace of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) = V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) holds
ex I being ( ( ) ( ) Basis of V : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) st I : ( ( ) ( ) Basis of b2 : ( ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) ( non empty V68() V105(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V106(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V107(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V108(b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) V115() V116() V117() ) VectSp of b1 : ( ( non empty non degenerated V68() V88() V94() V96() well-unital V102() V115() V116() V117() ) ( non empty non degenerated non trivial V68() V88() unital V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V115() V116() V117() ) Field) ) ) c= A : ( ( ) ( ) Subset of ) ;