:: VECTSP_4 semantic presentation

begin

definition
let GF be ( ( non empty ) ( non empty ) multMagma ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over GF : ( ( non empty ) ( non empty ) multMagma ) ) ;
let V1 be ( ( ) ( ) Subset of ) ;
attr V1 is linearly-closed means :: VECTSP_4:def 1
( ( for v, u being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in V1 : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in V1 : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) holds
v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) in V1 : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ) & ( for a being ( ( ) ( ) Element of ( ( ) ( ) set ) )
for v being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in V1 : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) in V1 : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ) );
end;

theorem :: VECTSP_4:1
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) & V1 : ( ( ) ( ) Subset of ) is linearly-closed holds
0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) ;

theorem :: VECTSP_4:2
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is linearly-closed holds
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) holds
- v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) ;

theorem :: VECTSP_4:3
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is linearly-closed holds
for v, u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) & u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) holds
v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) ;

theorem :: VECTSP_4:4
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) holds {(0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of bool the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is linearly-closed ;

theorem :: VECTSP_4:5
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for V1 being ( ( ) ( ) Subset of ) st the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) = V1 : ( ( ) ( ) Subset of ) holds
V1 : ( ( ) ( ) Subset of ) is linearly-closed ;

theorem :: VECTSP_4:6
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for V1, V2, V3 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is linearly-closed & V2 : ( ( ) ( ) Subset of ) is linearly-closed & V3 : ( ( ) ( ) Subset of ) = { (v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) where v, u is ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) & u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in V2 : ( ( ) ( ) Subset of ) ) } holds
V3 : ( ( ) ( ) Subset of ) is linearly-closed ;

theorem :: VECTSP_4:7
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for V1, V2 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is linearly-closed & V2 : ( ( ) ( ) Subset of ) is linearly-closed holds
V1 : ( ( ) ( ) Subset of ) /\ V2 : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is linearly-closed ;

definition
let GF be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;
mode Subspace of V -> ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) means :: VECTSP_4:def 2
( the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) c= the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) & 0. it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = 0. V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( V49(V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) & the addF of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V18([: the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like Function-like V18([: the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = the addF of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( Function-like V18([: the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) ) ( Relation-like Function-like V18([: the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) || the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) & the lmult of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V18([: the carrier of GF : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like Function-like V18([: the carrier of GF : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of GF : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = the lmult of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( Function-like V18([: the carrier of GF : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) ) ( Relation-like Function-like V18([: the carrier of GF : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of GF : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) | [: the carrier of GF : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of it : ( ( Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) VectSpStr over GF : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( Relation-like ) ( Relation-like Function-like ) set ) );
end;

theorem :: VECTSP_4:8
for x being ( ( ) ( ) set )
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st x : ( ( ) ( ) set ) in W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
x : ( ( ) ( ) set ) in W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:9
for x being ( ( ) ( ) set )
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st x : ( ( ) ( ) set ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
x : ( ( ) ( ) set ) in V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;

theorem :: VECTSP_4:10
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for w being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds w : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) is ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:11
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds 0. W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( V49(b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) right_complementable ) Element of the carrier of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:12
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds 0. W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( V49(b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) right_complementable ) Element of the carrier of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) = 0. W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( V49(b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) right_complementable ) Element of the carrier of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:13
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v, u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for w1, w2 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st w1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) & w2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds
w1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + w2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b5 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:14
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for a being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for w being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st w : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) * w : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b5 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:15
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for w being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st w : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds
- v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = - w : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:16
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v, u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for w1, w2 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st w1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) & w2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds
w1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - w2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b5 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:17
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds 0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:18
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds 0. W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( V49(b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) right_complementable ) Element of the carrier of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:19
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds 0. W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( V49(b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) right_complementable ) Element of the carrier of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) in V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;

theorem :: VECTSP_4:20
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u, v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:21
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for a being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:22
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
- v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:23
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u, v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:24
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) holds V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:25
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for X, V being ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) st V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of X : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & X : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) = X : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;

theorem :: VECTSP_4:26
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V, X, Y being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) st V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of X : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & X : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of Y : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of Y : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:27
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st the carrier of W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) c= the carrier of W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) holds
W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:28
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st ( for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

registration
let GF be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;
cluster non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed for ( ( ) ( ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;
end;

theorem :: VECTSP_4:29
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st the carrier of W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) = the carrier of W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) holds
W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) = W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:30
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st ( for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds
( v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) iff v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ) holds
W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) = W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:31
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st the carrier of W : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) = the carrier of V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) holds
W : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) = V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;

theorem :: VECTSP_4:32
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st ( for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
W : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) = V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;

theorem :: VECTSP_4:33
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for V1 being ( ( ) ( ) Subset of ) st the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) = V1 : ( ( ) ( ) Subset of ) holds
V1 : ( ( ) ( ) Subset of ) is linearly-closed ;

theorem :: VECTSP_4:34
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) & V1 : ( ( ) ( ) Subset of ) is linearly-closed holds
ex W being ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st V1 : ( ( ) ( ) Subset of ) = the carrier of W : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ;

definition
let GF be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;
func (0). V -> ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) means :: VECTSP_4:def 3
 the carrier of it : ( ( Function-like V18([:V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ( Relation-like Function-like V18([:V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) Element of bool [:[:V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = {(0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( V49(V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of bool the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let GF be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;
func (Omega). V -> ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) equals :: VECTSP_4:def 4
 VectSpStr(# the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the addF of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( Function-like V18([: the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like Function-like V18([: the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) , the lmult of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( Function-like V18([: the carrier of GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like Function-like V18([: the carrier of GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;
end;

theorem :: VECTSP_4:35
for x being ( ( ) ( ) set )
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) holds
( x : ( ( ) ( ) set ) in (0). V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) iff x : ( ( ) ( ) set ) = 0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( V49(b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP_4:36
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds (0). W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) = (0). V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:37
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds (0). W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) = (0). W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:38
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds (0). W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:39
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds (0). V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:40
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds (0). W1 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of W2 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:41
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) holds V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) is ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of (Omega). V : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

definition
let GF be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;
let v be ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;
let W be ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;
func v + W -> ( ( ) ( ) Subset of ) equals :: VECTSP_4:def 5
 { (v : ( ( Function-like V18([:V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ( Relation-like Function-like V18([:V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) Element of bool [:[:V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) + u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) where u is ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( ) Element of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) } ;
end;

definition
let GF be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;
let W be ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;
mode Coset of W -> ( ( ) ( ) Subset of ) means :: VECTSP_4:def 6
 ex v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st it : ( ( ) ( ) Element of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( Function-like V18([:V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ( Relation-like Function-like V18([:V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) Element of bool [:[:V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) ;
end;

theorem :: VECTSP_4:42
for x being ( ( ) ( ) set )
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( x : ( ( ) ( ) set ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) iff ex u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st
( u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & x : ( ( ) ( ) set ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: VECTSP_4:43
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( 0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) iff v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:44
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) ;

theorem :: VECTSP_4:45
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds (0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ;

theorem :: VECTSP_4:46
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + ((0). V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = {v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of bool the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:47
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + ((Omega). V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ;

theorem :: VECTSP_4:48
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( 0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) iff v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:49
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) iff v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:50
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for a being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
(a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ;

theorem :: VECTSP_4:51
for GF being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field)
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of GF : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) )
for a being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) st a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) <> 0. GF : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) & (a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) : ( ( ) ( non empty ) set ) holds
v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) ;

theorem :: VECTSP_4:52
for GF being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field)
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of GF : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) holds
( v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) iff (- v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:53
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u, v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) iff v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = (v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) ) ;

theorem :: VECTSP_4:54
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u, v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) iff v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = (v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) ) ;

theorem :: VECTSP_4:55
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v, u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) iff u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) ) ;

theorem :: VECTSP_4:56
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u, v1, v2 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) & u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in v2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) holds
v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = v2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) ;

theorem :: VECTSP_4:57
for GF being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field)
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of GF : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) )
for a being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) st a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) <> 1_ GF : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) & a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) : ( ( ) ( ) Subset of ) holds
v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ) ;

theorem :: VECTSP_4:58
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for a being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) ;

theorem :: VECTSP_4:59
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
- v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) ;

theorem :: VECTSP_4:60
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u, v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) iff u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:61
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v, u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) iff u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:62
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u, v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) iff ex v1 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st
( v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: VECTSP_4:63
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v1, v2 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( ex v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st
( v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) & v2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) ) iff v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - v2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:64
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v, u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) holds
ex v1 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st
( v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP_4:65
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v, u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) holds
ex v1 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st
( v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP_4:66
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W1, W2 being ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) iff W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) = W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:67
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v, u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W1, W2 being ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) = u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) holds
W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) = W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:68
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ex C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) st v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:69
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
( C : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) is linearly-closed iff C : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) = the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:70
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W1, W2 being ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for C1 being ( ( ) ( ) Coset of W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) )
for C2 being ( ( ) ( ) Coset of W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) st C1 : ( ( ) ( ) Coset of b3 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) = C2 : ( ( ) ( ) Coset of b4 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
W1 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) = W2 : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ;

theorem :: VECTSP_4:71
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds {v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of bool the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Coset of (0). V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:72
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is ( ( ) ( ) Coset of (0). V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
ex v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st V1 : ( ( ) ( ) Subset of ) = {v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of bool the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:73
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) is ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:74
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) holds the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) is ( ( ) ( ) Coset of (Omega). V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:75
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is ( ( ) ( ) Coset of (Omega). V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
V1 : ( ( ) ( ) Subset of ) = the carrier of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ;

theorem :: VECTSP_4:76
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
( 0. V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) iff C : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) = the carrier of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP_4:77
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
( u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) iff C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) = u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) : ( ( ) ( ) Subset of ) ) ;

theorem :: VECTSP_4:78
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u, v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) st u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b5 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) & v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b5 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
ex v1 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st
( v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP_4:79
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u, v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) st u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b5 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) & v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b5 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
ex v1 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) st
( v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) & u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP_4:80
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for v1, v2 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) holds
( ex C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) st
( v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b5 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) & v2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b5 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ) iff v1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - v2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:81
for GF being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for u being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) )
for B, C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) st u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in B : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) & u : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) holds
B : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) = C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) Subspace of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP_4:82
for GF being ( ( non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over GF : ( ( non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) )
for a, b being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds (a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - b : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = (a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) - (b : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over b1 : ( ( non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;