:: LMOD_5 semantic presentation

begin

definition
let R be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over R : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ;
let IT be ( ( ) ( ) Subset of ) ;
attr IT is linearly-independent means :: LMOD_5:def 1
 for l being ( ( ) ( Relation-like the carrier of V : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of R : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Linear_Combination of IT : ( ( ) ( ) BiModStr over R : ( ( ) ( ) 1-sorted ) ,V : ( ( ) ( ) 1-sorted ) ) ) st Sum l : ( ( ) ( Relation-like the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over R : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of R : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Linear_Combination of IT : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) = 0. V : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) holds
Carrier l : ( ( ) ( Relation-like the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over R : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of R : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Linear_Combination of IT : ( ( ) ( ) Subset of ) ) : ( ( finite ) ( finite ) Element of K19( the carrier of V : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) = {} : ( ( ) ( Function-like functional empty V18() V19() V20() V22() V23() V24() finite V29() ) set ) ;
end;

notation
let R be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over R : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ;
let IT be ( ( ) ( ) Subset of ) ;
antonym linearly-dependent IT for linearly-independent ;
end;

theorem :: LMOD_5:1
for R being ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) )
for A, B being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) c= B : ( ( ) ( ) Subset of ) & B : ( ( ) ( ) Subset of ) is linearly-independent holds
A : ( ( ) ( ) Subset of ) is linearly-independent ;

theorem :: LMOD_5:2
for R being ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) )
for A being ( ( ) ( ) Subset of ) st 0. R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( V49(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) Element of the carrier of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( non empty ) set ) ) <> 1. R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( non empty ) set ) ) & A : ( ( ) ( ) Subset of ) is linearly-independent holds
not 0. V : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) ) Element of the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) ) in A : ( ( ) ( ) Subset of ) ;

theorem :: LMOD_5:3
for R being ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) holds {} the carrier of V : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) : ( ( ) ( Function-like functional empty proper V18() V19() V20() V22() V23() V24() finite V29() ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) is linearly-independent ;

theorem :: LMOD_5:4
for R being ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) )
for v1, v2 being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) st 0. R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( V49(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) Element of the carrier of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( non empty ) set ) ) <> 1. R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( non empty ) set ) ) & {v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ,v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty finite ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) is linearly-independent holds
( v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) <> 0. V : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) ) Element of the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) ) & v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) <> 0. V : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) ) Element of the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LMOD_5:5
for R being ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) )
for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) st 0. R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( V49(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) Element of the carrier of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( non empty ) set ) ) <> 1. R : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) : ( ( ) ( non empty ) set ) ) holds
( {v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ,(0. V : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) ) Element of the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty finite ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) is linearly-dependent & {(0. V : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) ) ) Element of the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty finite ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V106(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V107(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V108(b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty right_complementable V94() well-unital V102() V112() V113() V114() ) ( non empty right_complementable V92() V94() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() ) Ring) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) is linearly-dependent ) ;

theorem :: LMOD_5:6
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for L being ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of V : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) )
for a being ( ( ) ( ) Scalar of ( ( ) ( non empty V12() ) set ) ) st a : ( ( ) ( ) Scalar of ( ( ) ( non empty V12() ) set ) ) <> 0. R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) ) holds
Carrier (a : ( ( ) ( ) Scalar of ( ( ) ( non empty V12() ) set ) ) * L : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ) : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) : ( ( finite ) ( finite ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = Carrier L : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) : ( ( finite ) ( finite ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LMOD_5:7
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for L being ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of V : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) )
for a being ( ( ) ( ) Scalar of ( ( ) ( non empty V12() ) set ) ) holds Sum (a : ( ( ) ( ) Scalar of ( ( ) ( non empty V12() ) set ) ) * L : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ) : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ) Scalar of ( ( ) ( non empty V12() ) set ) ) * (Sum L : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) ;

definition
let R be ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ;
let V be ( ( non empty right_complementable V105(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ;
let A be ( ( ) ( ) Subset of ) ;
func Lin A -> ( ( strict ) ( non empty right_complementable strict V105(R : ( ( ) ( ) 1-sorted ) ) V106(R : ( ( ) ( ) 1-sorted ) ) V107(R : ( ( ) ( ) 1-sorted ) ) V108(R : ( ( ) ( ) 1-sorted ) ) V112() V113() V114() ) Subspace of V : ( ( ) ( ) 1-sorted ) ) means :: LMOD_5:def 2
 the carrier of it : ( ( Function-like quasi_total ) ( Relation-like K20(A : ( ( ) ( ) BiModStr over R : ( ( ) ( ) 1-sorted ) ,V : ( ( ) ( ) 1-sorted ) ) ,A : ( ( ) ( ) BiModStr over R : ( ( ) ( ) 1-sorted ) ,V : ( ( ) ( ) 1-sorted ) ) ) : ( ( ) ( ) set ) -defined A : ( ( ) ( ) BiModStr over R : ( ( ) ( ) 1-sorted ) ,V : ( ( ) ( ) 1-sorted ) ) -valued Function-like quasi_total ) Element of K19(K20(K20(A : ( ( ) ( ) BiModStr over R : ( ( ) ( ) 1-sorted ) ,V : ( ( ) ( ) 1-sorted ) ) ,A : ( ( ) ( ) BiModStr over R : ( ( ) ( ) 1-sorted ) ,V : ( ( ) ( ) 1-sorted ) ) ) : ( ( ) ( ) set ) ,A : ( ( ) ( ) BiModStr over R : ( ( ) ( ) 1-sorted ) ,V : ( ( ) ( ) 1-sorted ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = { (Sum l : ( ( ) ( Relation-like the carrier of V : ( ( non empty right_complementable V105(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of A : ( ( ) ( ) Subset of ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) where l is ( ( ) ( Relation-like the carrier of V : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of R : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Linear_Combination of A : ( ( ) ( ) BiModStr over R : ( ( ) ( ) 1-sorted ) ,V : ( ( ) ( ) 1-sorted ) ) ) : verum } ;
end;

theorem :: LMOD_5:8
for x being ( ( ) ( ) set )
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for A being ( ( ) ( ) Subset of ) holds
( x : ( ( ) ( ) set ) in Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty right_complementable strict V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b3 : ( ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) iff ex l being ( ( ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of A : ( ( ) ( ) Subset of ) ) st x : ( ( ) ( ) set ) = Sum l : ( ( ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) -valued Function-like quasi_total ) Linear_Combination of b4 : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LMOD_5:9
for x being ( ( ) ( ) set )
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for A being ( ( ) ( ) Subset of ) st x : ( ( ) ( ) set ) in A : ( ( ) ( ) Subset of ) holds
x : ( ( ) ( ) set ) in Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty right_complementable strict V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b3 : ( ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b2 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ;

theorem :: LMOD_5:10
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) holds Lin ({} the carrier of V : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Function-like functional empty proper V18() V19() V20() V22() V23() V24() finite V29() ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) = (0). V : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ;

theorem :: LMOD_5:11
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for A being ( ( ) ( ) Subset of ) holds
( not Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) = (0). V : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) or A : ( ( ) ( ) Subset of ) = {} : ( ( ) ( Function-like functional empty V18() V19() V20() V22() V23() V24() finite V29() ) set ) or A : ( ( ) ( ) Subset of ) = {(0. V : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) : ( ( ) ( V49(b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ) Element of the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty finite ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LMOD_5:12
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for A being ( ( ) ( ) Subset of )
for W being ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of V : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) st 0. R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) ) <> 1. R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) ) & A : ( ( ) ( ) Subset of ) = the carrier of W : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) : ( ( ) ( non empty ) set ) holds
Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) = W : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ;

theorem :: LMOD_5:13
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for A being ( ( ) ( ) Subset of ) st 0. R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) ) <> 1. R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) : ( ( ) ( non empty V12() ) set ) ) & A : ( ( ) ( ) Subset of ) = the carrier of V : ( ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) holds
Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) = V : ( ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ;

theorem :: LMOD_5:14
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for A, B being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) c= B : ( ( ) ( ) Subset of ) holds
Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) is ( ( ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of Lin B : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ) ;

theorem :: LMOD_5:15
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for A, B being ( ( ) ( ) Subset of ) st Lin A : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) = V : ( ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) & A : ( ( ) ( ) Subset of ) c= B : ( ( ) ( ) Subset of ) holds
Lin B : ( ( ) ( ) Subset of ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) = V : ( ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ;

theorem :: LMOD_5:16
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for A, B being ( ( ) ( ) Subset of ) holds Lin (A : ( ( ) ( ) Subset of ) \/ B : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) = (Lin A : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) + (Lin B : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ;

theorem :: LMOD_5:17
for R being ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing)
for V being ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of R : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) )
for A, B being ( ( ) ( ) Subset of ) holds Lin (A : ( ( ) ( ) Subset of ) /\ B : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of K19( the carrier of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) is ( ( ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of (Lin A : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) /\ (Lin B : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) : ( ( strict ) ( non empty right_complementable strict V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) Subspace of b2 : ( ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) ( non empty right_complementable V105(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V106(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V107(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V108(b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) V112() V113() V114() ) LeftMod of b1 : ( ( non empty non degenerated right_complementable V94() V96() well-unital V102() V112() V113() V114() domRing-like ) ( non empty non degenerated V47() right_complementable V92() V94() V96() right-distributive left-distributive right_unital well-unital V102() left_unital V112() V113() V114() domRing-like ) domRing) ) ) ) ;