:: RUSUB_2 semantic presentation

begin

definition
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
let W1, W2 be ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;
func W1 + W2 -> ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( ) ( ) UNITSTR ) ) means :: RUSUB_2:def 1
 the carrier of it : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined V : ( ( ) ( ) UNITSTR ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = { (v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) UNITSTR ) : ( ( ) ( ) set ) ) where v, u is ( ( ) ( ) VECTOR of ( ( ) ( ) set ) ) : ( v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( ) Element of V : ( ( ) ( ) UNITSTR ) ) & u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W2 : ( ( Function-like V18([:V : ( ( ) ( ) UNITSTR ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) ) ) ( Relation-like [:V : ( ( ) ( ) UNITSTR ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined V : ( ( ) ( ) UNITSTR ) -valued Function-like V18([:V : ( ( ) ( ) UNITSTR ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) ) ) Element of bool [:[:V : ( ( ) ( ) UNITSTR ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

definition
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
let W1, W2 be ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;
func W1 /\ W2 -> ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( ) ( ) UNITSTR ) ) means :: RUSUB_2:def 2
 the carrier of it : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined V : ( ( ) ( ) UNITSTR ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = the carrier of W1 : ( ( ) ( ) Element of V : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) /\ the carrier of W2 : ( ( Function-like V18([:V : ( ( ) ( ) UNITSTR ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) ) ) ( Relation-like [:V : ( ( ) ( ) UNITSTR ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined V : ( ( ) ( ) UNITSTR ) -valued Function-like V18([:V : ( ( ) ( ) UNITSTR ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) ) ) Element of bool [:[:V : ( ( ) ( ) UNITSTR ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;
end;

begin

theorem :: RUSUB_2:1
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) iff ex v1, v2 being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st
( v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & x : ( ( ) ( ) set ) = v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: RUSUB_2:2
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st ( v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) or v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:3
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) iff ( x : ( ( ) ( ) set ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & x : ( ( ) ( ) set ) in W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ) ;

theorem :: RUSUB_2:4
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:5
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:6
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2, W3 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + (W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:7
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) & W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ) ;

theorem :: RUSUB_2:8
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for W2 being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) iff W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:9
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( ((0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + ((0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:10
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds
( ((0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = UNITSTR(# the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the ZeroF of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) , the U7 of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the scalar of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) UNITSTR ) & ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + ((0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = UNITSTR(# the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the ZeroF of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) , the U7 of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the scalar of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) UNITSTR ) ) ;

theorem :: RUSUB_2:11
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = UNITSTR(# the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the ZeroF of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) , the U7 of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the scalar of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) UNITSTR ) & W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = UNITSTR(# the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the ZeroF of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) , the U7 of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the scalar of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) UNITSTR ) ) ;

theorem :: RUSUB_2:12
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) holds ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) + ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) = V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ;

theorem :: RUSUB_2:13
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:14
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:15
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2, W3 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ (W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:16
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) & W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ) ;

theorem :: RUSUB_2:17
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for W1 being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( W1 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) iff W1 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W1 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:18
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( ((0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ ((0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:19
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds
( ((0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ ((0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:20
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:21
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) holds ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) /\ ((Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ) = V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) ;

theorem :: RUSUB_2:22
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:23
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for W2 being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:24
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for W2 being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ (W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:25
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2, W3 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + (W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:26
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2, W3 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + (W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:27
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2, W3 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ (W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:28
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2, W3 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ (W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:29
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2, W3 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + (W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:30
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( W1 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) iff W1 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = W1 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:31
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for W2, W3 being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W3 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W3 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:32
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( ( W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) or W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ) iff ex W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st the carrier of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) = the carrier of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) \/ the carrier of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) ;

begin

definition
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
func Subspaces V -> ( ( ) ( ) set ) means :: RUSUB_2:def 3
 for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in it : ( ( ) ( ) Element of V : ( ( ) ( ) UNITSTR ) ) iff x : ( ( ) ( ) set ) is ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( ) ( ) UNITSTR ) ) );
end;

registration
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
cluster Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( ) set ) -> non empty ;
end;

theorem :: RUSUB_2:33
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) holds V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) in Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ;

begin

definition
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
let W1, W2 be ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;
pred V is_the_direct_sum_of W1,W2 means :: RUSUB_2:def 4
( UNITSTR(# the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the ZeroF of V : ( ( ) ( ) set ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) , the U7 of V : ( ( ) ( ) set ) : ( ( Function-like V18([: the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like [: the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V18([: the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( ) ( ) set ) : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the scalar of V : ( ( ) ( ) set ) : ( ( Function-like V18([: the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [: the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([: the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[: the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) UNITSTR ) = W1 : ( ( ) ( ) set ) + W2 : ( ( Function-like V18([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) set ) ) ) ( Relation-like [:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined V : ( ( ) ( ) set ) -valued Function-like V18([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) set ) ) ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( ) ( ) set ) ) & W1 : ( ( ) ( ) set ) /\ W2 : ( ( Function-like V18([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) set ) ) ) ( Relation-like [:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined V : ( ( ) ( ) set ) -valued Function-like V18([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) set ) ) ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( ) ( ) set ) ) = (0). V : ( ( ) ( ) set ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( ) ( ) set ) ) );
end;

definition
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
let W be ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;
mode Linear_Compl of W -> ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( ) ( ) set ) ) means :: RUSUB_2:def 5
V : ( ( ) ( ) set ) is_the_direct_sum_of it : ( ( Function-like V18([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) set ) ) ) ( Relation-like [:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined V : ( ( ) ( ) set ) -valued Function-like V18([:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) set ) ) ) Element of bool [:[:V : ( ( ) ( ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ,W : ( ( ) ( ) set ) ;
end;

registration
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
let W be ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;
cluster non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like for ( ( ) ( ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) ) ;
end;

theorem :: RUSUB_2:34
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:35
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for L being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
( V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ,W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ) ;

begin

theorem :: RUSUB_2:36
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for L being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
( W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = UNITSTR(# the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the ZeroF of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) , the U7 of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the scalar of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) UNITSTR ) & L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) + W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = UNITSTR(# the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the ZeroF of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) , the U7 of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the scalar of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) UNITSTR ) ) ;

theorem :: RUSUB_2:37
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for L being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
( W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) /\ W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:38
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

theorem :: RUSUB_2:39
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds
( V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) , (Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of (Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) , (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:40
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for L being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ) ;

theorem :: RUSUB_2:41
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds
( (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of (Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) & (Omega). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of (0). V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ) ;

theorem :: RUSUB_2:42
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for C1 being ( ( ) ( ) Coset of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )
for C2 being ( ( ) ( ) Coset of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) st C1 : ( ( ) ( ) Coset of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) meets C2 : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
C1 : ( ( ) ( ) Coset of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) /\ C2 : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Coset of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:43
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) iff for C1 being ( ( ) ( ) Coset of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )
for C2 being ( ( ) ( ) Coset of W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ex v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st C1 : ( ( ) ( ) Coset of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) /\ C2 : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = {v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) } : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

begin

theorem :: RUSUB_2:44
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = UNITSTR(# the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the ZeroF of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) , the U7 of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the scalar of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) UNITSTR ) iff for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ex v1, v2 being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st
( v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: RUSUB_2:45
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for v, v1, v2, u1, u2 being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) & v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = u1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + u2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) & v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & u1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & u2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = u1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = u2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) ;

theorem :: RUSUB_2:46
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) = W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & ex v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st
for v1, v2, u1, u2 being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) & v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = u1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + u2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) & v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & u1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & u2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
( v1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = u1 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & v2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = u2 : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) holds
V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;

definition
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
let v be ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ;
let W1, W2 be ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;
assume V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ;
func v |-- (W1,W2) -> ( ( ) ( ) Element of [: the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) means :: RUSUB_2:def 6
( v : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) = (it : ( ( Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( ) Element of the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) + (it : ( ( Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( ) Element of the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) & it : ( ( Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( ) Element of the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) in W1 : ( ( Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) ) ( Relation-like [:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) -valued Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) ) Element of bool [:[:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & it : ( ( Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[:V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( ) Element of the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) in W2 : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) -defined V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) ,V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) );
end;

theorem :: RUSUB_2:47
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) )
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
(v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `1 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = (v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `2 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: RUSUB_2:48
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) )
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) is_the_direct_sum_of W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
(v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `2 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = (v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `1 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: RUSUB_2:49
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for L being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) )
for t being ( ( ) ( ) Element of [: the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) st (t : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `1) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) + (t : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `2) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) & t : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `1 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & t : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `2 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) in L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
t : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) ;

theorem :: RUSUB_2:50
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for L being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds ((v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `1) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) + ((v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `2) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ;

theorem :: RUSUB_2:51
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for L being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds
( (v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `1 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & (v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `2 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) in L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ) ;

theorem :: RUSUB_2:52
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for L being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds (v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `1 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = (v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ,W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `2 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: RUSUB_2:53
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for L being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds (v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ,L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `2 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = (v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) |-- (L : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Linear_Compl of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ,W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) )) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) `1 : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ;

begin

definition
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
func SubJoin V -> ( ( Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) means :: RUSUB_2:def 7
 for A1, A2 being ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) )
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) st A1 : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & A2 : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
it : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) . (A1 : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ,A2 : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) = W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) + W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) ;
end;

definition
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
func SubMeet V -> ( ( Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) means :: RUSUB_2:def 8
 for A1, A2 being ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) )
for W1, W2 being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) st A1 : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & A2 : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) = W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) holds
it : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) . (A1 : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ,A2 : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) = W1 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) ;
end;

begin

theorem :: RUSUB_2:54
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds LattStr(# (Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(SubJoin V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ,(SubMeet V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) LattStr ) is ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;

registration
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
cluster LattStr(# (Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(SubJoin V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ,(SubMeet V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) LattStr ) -> strict Lattice-like ;
end;

theorem :: RUSUB_2:55
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds LattStr(# (Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(SubJoin V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ,(SubMeet V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is lower-bounded ;

theorem :: RUSUB_2:56
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds LattStr(# (Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(SubJoin V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ,(SubMeet V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is upper-bounded ;

theorem :: RUSUB_2:57
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds LattStr(# (Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(SubJoin V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ,(SubMeet V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is ( ( non empty Lattice-like V77() ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded V77() ) 01_Lattice) ;

theorem :: RUSUB_2:58
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds LattStr(# (Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(SubJoin V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ,(SubMeet V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is modular ;

theorem :: RUSUB_2:59
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) holds LattStr(# (Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(SubJoin V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ,(SubMeet V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) ,(Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is complemented ;

registration
let V be ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ;
cluster LattStr(# (Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(SubJoin V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ,(SubMeet V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) -valued Function-like V18([:(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) ,(Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) ) BinOp of Subspaces V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) UNITSTR ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -> strict modular lower-bounded upper-bounded complemented ;
end;

theorem :: RUSUB_2:60
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W1, W2, W3 being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st W1 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
W1 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) is ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of W2 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) /\ W3 : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

begin

theorem :: RUSUB_2:61
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) st ( for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) holds
W : ( ( strict ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) = UNITSTR(# the carrier of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the ZeroF of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) , the U7 of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:K93() : ( ( ) ( V33() V34() V35() V39() ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) , the scalar of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined K93() : ( ( ) ( V33() V34() V35() V39() ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,K93() : ( ( ) ( V33() V34() V35() V39() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) UNITSTR ) ;

theorem :: RUSUB_2:62
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ex C being ( ( ) ( ) Coset of W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) st v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b2 : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) ) ;

theorem :: RUSUB_2:63
for V being ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace)
for W being ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of V : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) )
for v being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) iff ex u being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) st
( u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) Subspace of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) ) & x : ( ( ) ( ) set ) = v : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + u : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( non empty ) set ) ) ) ) ;