:: VECTSP10 semantic presentation

begin

definition
let K be ( ( ) ( ) doubleLoopStr ) ;
func StructVectSp K -> ( ( strict ) ( strict ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) equals :: VECTSP10:def 1
 VectSpStr(# the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the addF of K : ( ( ) ( ) 1-sorted ) : ( ( Function-like quasi_total ) ( V1() V4([: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ,(0. K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) , the multF of K : ( ( ) ( ) 1-sorted ) : ( ( Function-like quasi_total ) ( V1() V4([: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
end;

registration
let K be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
cluster StructVectSp K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( strict ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) -> non empty strict ;
end;

registration
let K be ( ( non empty Abelian ) ( non empty Abelian ) doubleLoopStr ) ;
cluster StructVectSp K : ( ( non empty Abelian ) ( non empty Abelian ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) VectSpStr over K : ( ( non empty Abelian ) ( non empty Abelian ) doubleLoopStr ) ) -> Abelian strict ;
end;

registration
let K be ( ( non empty add-associative ) ( non empty add-associative ) doubleLoopStr ) ;
cluster StructVectSp K : ( ( non empty add-associative ) ( non empty add-associative ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) VectSpStr over K : ( ( non empty add-associative ) ( non empty add-associative ) doubleLoopStr ) ) -> add-associative strict ;
end;

registration
let K be ( ( non empty right_zeroed ) ( non empty right_zeroed ) doubleLoopStr ) ;
cluster StructVectSp K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) doubleLoopStr ) ) -> right_zeroed strict ;
end;

registration
let K be ( ( non empty right_complementable ) ( non empty right_complementable ) doubleLoopStr ) ;
cluster StructVectSp K : ( ( non empty right_complementable ) ( non empty right_complementable ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) VectSpStr over K : ( ( non empty right_complementable ) ( non empty right_complementable ) doubleLoopStr ) ) -> right_complementable strict ;
end;

registration
let K be ( ( non empty associative distributive left_unital ) ( non empty associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ;
cluster StructVectSp K : ( ( non empty associative distributive left_unital ) ( non empty associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) VectSpStr over K : ( ( non empty associative distributive left_unital ) ( non empty associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) -> strict vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

registration
let K be ( ( non empty non degenerated ) ( non empty non degenerated non trivial ) doubleLoopStr ) ;
cluster StructVectSp K : ( ( non empty non degenerated ) ( non empty non degenerated non trivial ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) VectSpStr over K : ( ( non empty non degenerated ) ( non empty non degenerated non trivial ) doubleLoopStr ) ) -> non trivial strict ;
end;

registration
let K be ( ( non empty non degenerated ) ( non empty non degenerated non trivial ) doubleLoopStr ) ;
cluster non trivial for ( ( ) ( ) VectSpStr over K : ( ( non empty non degenerated ) ( non empty non degenerated non trivial ) doubleLoopStr ) ) ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ;
cluster non empty right_complementable add-associative right_zeroed strict for ( ( ) ( ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ;
cluster non empty right_complementable add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital for ( ( ) ( ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) ;
end;

registration
let K be ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ;
cluster non empty non trivial right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital for ( ( ) ( ) VectSpStr over K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) ;
end;

theorem :: VECTSP10:1
for K being ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr )
for a being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) )
for V being ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) holds
( (0. K : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( V48(b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( V48(b3 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) & a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) * (0. V : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( V48(b3 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( V48(b3 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative distributive left_unital ) ( non empty right_complementable add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP10:2
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for S, T being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st S : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) /\ T : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) = (0). V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) & v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in S : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) & v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in T : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( V48(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP10:3
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for x being ( ( ) ( ) set )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) holds
( x : ( ( ) ( ) set ) in Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) iff ex a being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) st x : ( ( ) ( ) set ) = a : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP10:4
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for a, b being ( ( ) ( right_complementable ) Scalar of ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) <> 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( V48(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) & a : ( ( ) ( right_complementable ) Scalar of ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) = b : ( ( ) ( right_complementable ) Scalar of ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( right_complementable ) Scalar of ) = b : ( ( ) ( right_complementable ) Scalar of ) ;

theorem :: VECTSP10:5
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) st V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) is_the_direct_sum_of W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
for v, v1, v2 being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) & v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) & v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) holds
v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

theorem :: VECTSP10:6
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) st V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) is_the_direct_sum_of W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
for v, v1, v2 being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) holds
v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP10:7
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) st V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) is_the_direct_sum_of W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
for v, v1, v2 being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) holds
( v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) & v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) ;

theorem :: VECTSP10:8
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) st V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) is_the_direct_sum_of W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
for v, v1, v2 being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) holds
v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

theorem :: VECTSP10:9
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) st V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) is_the_direct_sum_of W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,(0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( V48(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

theorem :: VECTSP10:10
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) st V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) is_the_direct_sum_of W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [(0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( V48(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

theorem :: VECTSP10:11
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for V1 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for W1 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) holds
v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) is ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP10:12
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for V1, V2, W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) st W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b5 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) = V1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) & W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b5 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) = V2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b5 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) + W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b5 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b5 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) = V1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) + V2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;

theorem :: VECTSP10:13
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for w being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) holds
Lin {w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) = Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ;

theorem :: VECTSP10:14
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for X being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) st not v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) holds
for y being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) & W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) = X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) holds
X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) is_the_direct_sum_of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) , Lin {y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ;

theorem :: VECTSP10:15
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for X being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) )
for y being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) & X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) = W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) & not v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) holds
y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ,(Lin {y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) : ( ( ) ( ) Element of [: the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [(0. W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) : ( ( ) ( V48(b6 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) right_complementable ) Element of the carrier of b6 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b6 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( ) ( non empty ) set ) , the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

theorem :: VECTSP10:16
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for X being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) )
for y being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) & X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) = W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) & not v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) holds
for w being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) holds
w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ,(Lin {y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) : ( ( ) ( ) Element of [: the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,(0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( V48(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

theorem :: VECTSP10:17
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ex v1, v2 being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

theorem :: VECTSP10:18
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for X being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) )
for y being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) & X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) = W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) & not v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) holds
for w being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ex x being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ex r being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) st w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ,(Lin {y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) : ( ( ) ( ) Element of [: the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [x : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,(r : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

theorem :: VECTSP10:19
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for X being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) )
for y being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) & X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) = W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) & not v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) holds
for w1, w2 being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for x1, x2 being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for r1, r2 being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) st w1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ,(Lin {y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) : ( ( ) ( ) Element of [: the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [x1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,(r1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) & w2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ,(Lin {y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) : ( ( ) ( ) Element of [: the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [x2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,(r2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) holds
(w1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + w2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ,(Lin {y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) : ( ( ) ( ) Element of [: the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [(x1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + x2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) ,((r1 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) + r2 : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

theorem :: VECTSP10:20
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for X being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) )
for y being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) & X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) = W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) & not v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) holds
for w being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for t, r being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) st w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ,(Lin {y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) : ( ( ) ( ) Element of [: the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [x : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ,(r : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) holds
(t : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ,(Lin {y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ) : ( ( ) ( ) Element of [: the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of (b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) = [(t : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * x : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) ,((t : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * r : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non empty ) Element of [: the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) ) ;

begin

definition
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let W be ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func CosetSet (V,W) -> ( ( non empty ) ( non empty ) Subset-Family of ) equals :: VECTSP10:def 2
 { A : ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) where A is ( ( ) ( ) Coset of W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : verum } ;
end;

definition
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let W be ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func addCoset (V,W) -> ( ( Function-like quasi_total ) ( V1() V4([:(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) ) V5( CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty V14([:(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) ) quasi_total ) BinOp of CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) means :: VECTSP10:def 3
 for A, B being ( ( ) ( ) Element of CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) )
for a, b being ( ( ) ( right_complementable ) Vector of ( ( ) ( ) set ) ) st A : ( ( ) ( ) Element of CosetSet (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = a : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & B : ( ( ) ( ) Element of CosetSet (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = b : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) . (A : ( ( ) ( ) Element of CosetSet (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ,B : ( ( ) ( ) Element of CosetSet (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) : ( ( ) ( ) Element of CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = (a : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + b : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) + W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let W be ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func zeroCoset (V,W) -> ( ( ) ( ) Element of CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) equals :: VECTSP10:def 4
 the carrier of W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ;
end;

definition
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let W be ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func lmultCoset (V,W) -> ( ( Function-like quasi_total ) ( V1() V4([: the carrier of K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) ) V5( CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty V14([: the carrier of K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) ) quasi_total ) Function of [: the carrier of K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) , CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) means :: VECTSP10:def 5
 for z being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) )
for A being ( ( ) ( ) Element of CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) )
for a being ( ( ) ( right_complementable ) Vector of ( ( ) ( ) set ) ) st A : ( ( ) ( ) Element of CosetSet (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = a : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) . (z : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Element of CosetSet (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) : ( ( ) ( ) Element of CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = (z : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) * a : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) + W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let W be ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func VectQuot (V,W) -> ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) ) means :: VECTSP10:def 6
( the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) & the addF of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V4([: the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5( the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[: the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) , the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) = addCoset (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V4([:(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) ) V5( CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty V14([:(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) ) quasi_total ) BinOp of CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) & 0. it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = zeroCoset (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) & the lmult of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V4([: the carrier of K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) , the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5( the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) , the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) , the carrier of it : ( ( ) ( ) Element of V : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) = lmultCoset (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V4([: the carrier of K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) ) V5( CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty V14([: the carrier of K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) ) quasi_total ) Function of [: the carrier of K : ( ( non empty non degenerated right_complementable Abelian add-associative right_zeroed associative distributive left_unital ) ( non empty non degenerated non trivial right_complementable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,(CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty ) ( non empty ) Subset-Family of ) :] : ( ( ) ( V1() non empty ) set ) , CosetSet (V : ( ( ) ( ) set ) ,W : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) );
end;

theorem :: VECTSP10:21
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
( zeroCoset (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of CosetSet (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) = (0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( V48(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & 0. (VectQuot (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( V48( VectQuot (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of (VectQuot (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = zeroCoset (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of CosetSet (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ;

theorem :: VECTSP10:22
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for w being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) holds
( w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) & ex v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP10:23
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) holds
( v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) & v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) ;

theorem :: VECTSP10:24
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for A being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) holds A : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) is ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP10:25
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for A being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( right_complementable ) Scalar of ) st A : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( right_complementable ) Scalar of ) * A : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of (VectQuot (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = (a : ( ( ) ( right_complementable ) Scalar of ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: VECTSP10:26
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for A1, A2 being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for v1, v2 being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st A1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & A2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
A1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + A2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of (VectQuot (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = (v1 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + v2 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

begin

theorem :: VECTSP10:27
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for X being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) )
for fi being ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for y being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) & not v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) holds
for r being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ex psi being ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of (b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of (b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st
( psi : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of (b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of (b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) | the carrier of X : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( V1() V4( the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) V4( the carrier of (b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like ) Element of bool [: the carrier of (b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) = fi : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) & psi : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of (b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of (b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) + (Lin {b5 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) . y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = r : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) ;

registration
let K be ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ;
cluster V1() V4( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive 0-preserving for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ;
let V be ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) ;
cluster Function-like quasi_total additive -> Function-like quasi_total 0-preserving for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster Function-like quasi_total homogeneous -> Function-like quasi_total 0-preserving for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let K be ( ( non empty ) ( non empty ) ZeroStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ;
cluster 0Functional V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( Function-like quasi_total ) ( V1() V4( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) -> Function-like constant quasi_total ;
end;

registration
let K be ( ( non empty ) ( non empty ) ZeroStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ;
cluster V1() V4( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) Function-like constant non empty V14( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let K be ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ;
let V be ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) ;
let f be ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total 0-preserving ) Functional of ( ( ) ( non empty ) set ) ) ;
:: original: constant
redefine attr f is constant means :: VECTSP10:def 7
f : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ,V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ,V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) :] : ( ( ) ( V1() ) set ) ,V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) = 0Functional V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( Function-like quasi_total ) ( V1() V4( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) Function-like constant non empty V14( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ;
let V be ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) ;
cluster V1() V4( the carrier of V : ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like constant non empty V14( the carrier of V : ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive 0-preserving for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ;
let V be ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ;
cluster V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non constant non empty non trivial V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ;
let V be ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ;
cluster Function-like trivial quasi_total -> Function-like constant quasi_total for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let K be ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ;
let V be ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ;
let v be ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) ;
let W be ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Linear_Compl of Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) ;
assume v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) <> 0. V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( V48(V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) right_complementable ) Element of the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) ;
func coeffFunctional (v,W) -> ( ( Function-like V8() non trivial quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) means :: VECTSP10:def 8
( it : ( ( Function-like quasi_total ) ( V1() V4([: the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) . v : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = 1_ K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) & it : ( ( Function-like quasi_total ) ( V1() V4([: the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) | the carrier of W : ( ( ) ( ) Element of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) set ) : ( ( ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V4( the carrier of W : ( ( ) ( ) Element of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) set ) ) V4([: the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) = 0Functional W : ( ( ) ( ) Element of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( Function-like quasi_total ) ( V1() V4( the carrier of W : ( ( ) ( ) Element of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V14( the carrier of W : ( ( ) ( ) Element of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) set ) ) quasi_total ) Element of bool [: the carrier of W : ( ( ) ( ) Element of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) );
end;

theorem :: VECTSP10:28
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for f being ( ( Function-like V8() quasi_total 0-preserving ) ( V1() V4( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Functional of ( ( ) ( non empty non trivial ) set ) ) ex v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) st
( v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) <> 0. V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( V48(b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) & f : ( ( Function-like V8() quasi_total 0-preserving ) ( V1() V4( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Functional of ( ( ) ( non empty non trivial ) set ) ) . v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) <> 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( V48(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: VECTSP10:29
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) )
for a being ( ( ) ( right_complementable ) Scalar of )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Linear_Compl of Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) <> 0. V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( V48(b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) holds
(coeffFunctional (v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Linear_Compl of Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) )) : ( ( Function-like V8() non trivial quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) . (a : ( ( ) ( right_complementable ) Scalar of ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( right_complementable ) Scalar of ) ;

theorem :: VECTSP10:30
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v, w being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Linear_Compl of Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) <> 0. V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( V48(b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) & w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Linear_Compl of Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) holds
(coeffFunctional (v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Linear_Compl of Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) )) : ( ( Function-like V8() non trivial quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) . w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( V48(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: VECTSP10:31
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field)
for V being ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) )
for v, w being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) )
for a being ( ( ) ( right_complementable ) Scalar of )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Linear_Compl of Lin {v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) st v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) <> 0. V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( V48(b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) right_complementable ) Element of the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) & w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Linear_Compl of Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) holds
(coeffFunctional (v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Linear_Compl of Lin {b3 : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ) )) : ( ( Function-like V8() non trivial quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) . ((a : ( ( ) ( right_complementable ) Scalar of ) * v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) + w : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( right_complementable ) Scalar of ) ;

theorem :: VECTSP10:32
for K being ( ( non empty ) ( non empty ) addLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) )
for f, g being ( ( Function-like quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Functional of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds (f : ( ( Function-like quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Functional of ( ( ) ( non empty ) set ) ) - g : ( ( Function-like quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Functional of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) . v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Functional of ( ( ) ( non empty ) set ) ) . v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) - (g : ( ( Function-like quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Functional of ( ( ) ( non empty ) set ) ) . v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ;
let V be ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ;
cluster V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) *' : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) -> non empty non trivial strict ;
end;

begin

definition
let K be ( ( non empty ) ( non empty ) ZeroStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ;
let f be ( ( Function-like quasi_total ) ( V1() V4( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Functional of ( ( ) ( non empty ) set ) ) ;
func ker f -> ( ( ) ( ) Subset of ) equals :: VECTSP10:def 9
 { v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) where v is ( ( ) ( ) Vector of ( ( ) ( non empty non trivial ) set ) ) : f : ( ( Function-like quasi_total ) ( V1() V4([:V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) Function-like quasi_total ) Element of bool [:[:V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) . v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( V48(K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) right_complementable ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) } ;
end;

registration
let K be ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ;
let f be ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total 0-preserving ) Functional of ( ( ) ( non empty ) set ) ) ;
cluster ker f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) -> non empty ;
end;

theorem :: VECTSP10:33
for K being ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for f being ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds ker f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) Subset of ) is linearly-closed ;

definition
let K be ( ( non empty ) ( non empty ) ZeroStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ;
let f be ( ( Function-like quasi_total ) ( V1() V4( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Functional of ( ( ) ( non empty ) set ) ) ;
attr f is degenerated means :: VECTSP10:def 10
 ker f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Subset of ) <> {(0. V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( V48(V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) right_complementable ) Element of the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) } : ( ( ) ( trivial V29() ) Element of bool the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let K be ( ( non empty non degenerated ) ( non empty non degenerated non trivial ) doubleLoopStr ) ;
let V be ( ( non trivial ) ( non empty non trivial ) VectSpStr over K : ( ( non empty non degenerated ) ( non empty non degenerated non trivial ) doubleLoopStr ) ) ;
cluster Function-like quasi_total 0-preserving non degenerated -> Function-like non constant quasi_total for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let f be ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func Ker f -> ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) means :: VECTSP10:def 11
 the carrier of it : ( ( ) ( ) Element of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) set ) = ker f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Subset of ) ;
end;

definition
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let W be ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
let f be ( ( Function-like quasi_total additive ) ( V1() V4( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive 0-preserving ) Functional of ( ( ) ( non empty ) set ) ) ;
assume the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) c= ker f : ( ( Function-like quasi_total additive ) ( V1() V4( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive 0-preserving ) Functional of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Subset of ) ;
func QFunctional (f,W) -> ( ( Function-like quasi_total additive ) ( V1() V4( the carrier of (VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of (VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,W : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive 0-preserving ) Functional of ( ( ) ( non empty ) set ) ) means :: VECTSP10:def 12
 for A being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty non trivial ) set ) ) st A : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + W : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( Function-like quasi_total ) ( V1() V4([: the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ) V5(V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) ,V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( non empty ) set ) ) . A : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = f : ( ( ) ( ) Element of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . a : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ;
end;

theorem :: VECTSP10:34
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for f being ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) st the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) c= ker f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) Subset of ) holds
QFunctional (f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ,W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( Function-like quasi_total additive ) ( V1() V4( the carrier of (VectQuot (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of (VectQuot (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive 0-preserving ) Functional of ( ( ) ( non empty ) set ) ) is homogeneous ;

definition
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let f be ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func CQFunctional f -> ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of (VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of (VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) equals :: VECTSP10:def 13
 QFunctional (f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Ker f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( Function-like quasi_total additive ) ( V1() V4( the carrier of (VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of (VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like quasi_total 0-preserving ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive 0-preserving ) Functional of ( ( ) ( non empty ) set ) ) ;
end;

theorem :: VECTSP10:35
for K being ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for f being ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for A being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) + (Ker f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(CQFunctional f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of (VectQuot (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker b3 : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of (VectQuot (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker b3 : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of VectQuot (b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker b3 : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . A : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . v : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ;
let V be ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ;
let f be ( ( Function-like V8() quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) Field) ) ) ;
cluster CQFunctional f : ( ( Function-like V8() quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of (VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like V8() quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like non empty V14( the carrier of (VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like V8() quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of VectQuot (V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like V8() quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) V5( the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V8() non empty non trivial V14( the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) ) quasi_total additive homogeneous 0-preserving ) Element of bool [: the carrier of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty non trivial ) set ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive ) ( non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) -> Function-like V8() quasi_total additive homogeneous ;
end;

registration
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let f be ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
cluster CQFunctional f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) Element of bool [: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of (VectQuot (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) Element of bool [: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of (VectQuot (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) Element of bool [: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) linear-Functional of VectQuot (V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ,(Ker f : ( ( Function-like quasi_total additive homogeneous ) ( V1() V4( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like non empty V14( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous 0-preserving ) Element of bool [: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( V1() non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) -> Function-like quasi_total additive homogeneous non degenerated ;
end;