:: MOD_2 semantic presentation

begin

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
func TrivialLMod R -> ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( ) ( ) GroupMorphismStr ) ) equals :: MOD_2:def 1
 VectSpStr(# 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,op2 : ( ( V6() quasi_total ) ( V1() V4([:1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( non empty ) set ) ) V5(1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ) V6() non empty V14([:1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[:1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( non empty ) set ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,op0 : ( ( ) ( empty ) Element of 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ) ,(pr2 ( the carrier of R : ( ( ) ( ) GroupMorphismStr ) : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( V6() quasi_total ) ( V1() V4([: the carrier of R : ( ( ) ( ) GroupMorphismStr ) : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( ) set ) ) V5(1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ) V6() V14([: the carrier of R : ( ( ) ( ) GroupMorphismStr ) : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( ) set ) ) quasi_total ) Element of bool [:[: the carrier of R : ( ( ) ( ) GroupMorphismStr ) : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) VectSpStr over R : ( ( ) ( ) GroupMorphismStr ) ) ;
end;

theorem :: MOD_2:1
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for x being ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Vector of ( ( ) ( non empty ) set ) ) holds x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Vector of ( ( ) ( non empty ) set ) ) = 0. (TrivialLMod R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( V46( TrivialLMod b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of (TrivialLMod b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) ;

definition
let R be ( ( non empty ) ( non empty ) multMagma ) ;
let G, H be ( ( non empty ) ( non empty ) VectSpStr over R : ( ( non empty ) ( non empty ) multMagma ) ) ;
let f be ( ( V6() quasi_total ) ( V1() V4( the carrier of G : ( ( non empty ) ( non empty ) VectSpStr over R : ( ( non empty ) ( non empty ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of H : ( ( non empty ) ( non empty ) VectSpStr over R : ( ( non empty ) ( non empty ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of G : ( ( non empty ) ( non empty ) VectSpStr over R : ( ( non empty ) ( non empty ) multMagma ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
attr f is homogeneous means :: MOD_2:def 2
 for a being ( ( ) ( ) Scalar of ( ( ) ( ) set ) )
for x being ( ( ) ( ) Vector of ( ( ) ( ) set ) ) holds f : ( ( V6() quasi_total ) ( V1() V4([:H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) ,H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) ) V6() quasi_total ) Element of bool [:[:H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) ,H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . (a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * x : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of G : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) = a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * (f : ( ( V6() quasi_total ) ( V1() V4([:H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) ,H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) ) V6() quasi_total ) Element of bool [:[:H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) ,H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of H : ( ( ) ( ) BiModStr over R : ( ( ) ( ) GroupMorphismStr ) ,G : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: MOD_2:2
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G, H, S being ( ( non empty ) ( non empty ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for f being ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( V6() quasi_total ) ( V1() V4( the carrier of b3 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b4 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b3 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is homogeneous & g : ( ( V6() quasi_total ) ( V1() V4( the carrier of b3 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b4 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b3 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is homogeneous holds
g : ( ( V6() quasi_total ) ( V1() V4( the carrier of b3 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b4 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b3 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( V6() ) ( V1() V4( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b4 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is homogeneous ;

registration
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G, H be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
cluster ZeroMap (G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Element of bool [: the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -> V6() quasi_total homogeneous ;
end;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
attr c2 is strict ;
struct LModMorphismStr over R -> ;
aggr LModMorphismStr(# Dom, Cod, Fun #) -> ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
sel Dom c2 -> ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
sel Cod c2 -> ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
sel Fun c2 -> ( ( V6() quasi_total ) ( V1() V4( the carrier of the Dom of c2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of the Cod of c2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of the Dom of c2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
end;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let f be ( ( ) ( ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
func dom f -> ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) equals :: MOD_2:def 3
 the Dom of f : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
func cod f -> ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) equals :: MOD_2:def 4
 the Cod of f : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
end;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let f be ( ( ) ( ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
func fun f -> ( ( V6() quasi_total ) ( V1() V4( the carrier of (dom f : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (cod f : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of (dom f : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) equals :: MOD_2:def 5
 the Fun of f : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of the Dom of f : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of the Cod of f : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of the Dom of f : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
end;

theorem :: MOD_2:3
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G1, G2 being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for f being ( ( ) ( ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for f0 being ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = LModMorphismStr(# G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,f0 : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
( dom f : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & cod f : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & fun f : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of (dom b4 : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (cod b4 : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of (dom b4 : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f0 : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G, H be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
func ZERO (G,H) -> ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) equals :: MOD_2:def 6
 LModMorphismStr(# G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ,H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) ,(ZeroMap (G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ,H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) )) : ( ( V6() quasi_total ) ( V1() V4( the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( ) ( non empty ) set ) ) V5( the carrier of H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( ) ( ) set ) ) V6() non empty V14( the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
end;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let IT be ( ( ) ( ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
attr IT is LModMorphism-like means :: MOD_2:def 7
( fun IT : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of (dom IT : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (cod IT : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of (dom IT : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is additive & fun IT : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of (dom IT : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (cod IT : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of (dom IT : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is homogeneous );
end;

registration
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
cluster strict LModMorphism-like for ( ( ) ( ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
end;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
mode LModMorphism of R is ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
end;

theorem :: MOD_2:4
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for F being ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
( the Fun of F : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of the Dom of b2 : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of the Cod of b2 : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of the Dom of b2 : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is additive & the Fun of F : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of the Dom of b2 : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of the Cod of b2 : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of the Dom of b2 : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is homogeneous ) ;

registration
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G, H be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
cluster ZERO (G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) : ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) -> strict LModMorphism-like ;
end;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G, H be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
mode Morphism of G,H -> ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) means :: MOD_2:def 8
( dom it : ( ( V6() quasi_total ) ( V1() V4([:H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) ,H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) :] : ( ( ) ( ) set ) ) V5(H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) ) V6() quasi_total ) Element of bool [:[:H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) ,H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) :] : ( ( ) ( ) set ) ,H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) & cod it : ( ( V6() quasi_total ) ( V1() V4([:H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) ,H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) :] : ( ( ) ( ) set ) ) V5(H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) ) V6() quasi_total ) Element of bool [:[:H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) ,H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) :] : ( ( ) ( ) set ) ,H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = H : ( ( ) ( ) GroupMorphism_DOMAIN of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ,G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V162() V163() V164() V165() ) addLoopStr ) ) );
end;

registration
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G, H be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
cluster strict LModMorphism-like for ( ( ) ( ) Morphism of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) ;
end;

theorem :: MOD_2:5
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G, H being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for f being ( ( ) ( ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) st dom f : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & cod f : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & fun f : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of (dom b4 : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (cod b4 : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of (dom b4 : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is additive & fun f : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of (dom b4 : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (cod b4 : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of (dom b4 : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is homogeneous holds
f : ( ( ) ( ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) is ( ( ) ( LModMorphism-like ) Morphism of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ;

theorem :: MOD_2:6
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G, H being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for f being ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is additive & f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is homogeneous holds
LModMorphismStr(# G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) is ( ( strict ) ( strict LModMorphism-like ) Morphism of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ;

registration
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
cluster id G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Element of bool [: the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -> V6() quasi_total homogeneous ;
end;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
func ID G -> ( ( strict ) ( strict LModMorphism-like ) Morphism of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) equals :: MOD_2:def 9
 LModMorphismStr(# G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,(id G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total additive homogeneous ) Element of bool [: the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
end;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G, H be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
:: original: ZERO
redefine func ZERO (G,H) -> ( ( strict ) ( strict LModMorphism-like ) Morphism of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) ;
end;

theorem :: MOD_2:7
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G, H being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for F being ( ( ) ( LModMorphism-like ) Morphism of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ex f being ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st
( LModMorphismStr(# the Dom of F : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) , the Cod of F : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) , the Fun of F : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of the Dom of b4 : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of the Cod of b4 : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of the Dom of b4 : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = LModMorphismStr(# G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is additive & f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is homogeneous ) ;

theorem :: MOD_2:8
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G, H being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for F being ( ( strict ) ( strict LModMorphism-like ) Morphism of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ex f being ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st F : ( ( strict ) ( strict LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) = LModMorphismStr(# G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;

theorem :: MOD_2:9
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for F being ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ex G, H being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) st F : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) is ( ( ) ( LModMorphism-like ) Morphism of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ;

theorem :: MOD_2:10
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for F being ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ex G, H being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ex f being ( ( V6() quasi_total ) ( V1() V4( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st
( F : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) is ( ( strict ) ( strict LModMorphism-like ) Morphism of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) & F : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = LModMorphismStr(# G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,H : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is additive & f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is homogeneous ) ;

theorem :: MOD_2:11
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for g, f being ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) st dom g : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = cod f : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
ex G1, G2, G3 being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) st
( g : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) is ( ( ) ( LModMorphism-like ) Morphism of G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) & f : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) is ( ( ) ( LModMorphism-like ) Morphism of G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ) ;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G, F be ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
assume dom G : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = cod F : ( ( LModMorphism-like ) ( LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
func G * F -> ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) means :: MOD_2:def 10
 for G1, G2, G3 being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) )
for g being ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( V6() quasi_total ) ( V1() V4( the carrier of b1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st LModMorphismStr(# the Dom of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) , the Cod of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) , the Fun of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of the Dom of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of the Cod of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of the Dom of G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = LModMorphismStr(# G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,g : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) & LModMorphismStr(# the Dom of F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) , the Cod of F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) , the Fun of F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of the Dom of F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of the Cod of F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of the Dom of F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = LModMorphismStr(# G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) holds
it : ( ( V6() quasi_total ) ( V1() V4([:F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) :] : ( ( ) ( ) set ) ) V5(F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) V6() quasi_total ) Element of bool [:[:F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) :] : ( ( ) ( ) set ) ,F : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = LModMorphismStr(# G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,(g : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( V6() ) ( V1() V4( the carrier of b1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of b1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
end;

theorem :: MOD_2:12
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G2, G3, G1 being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for G being ( ( ) ( LModMorphism-like ) Morphism of G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) )
for F being ( ( ) ( LModMorphism-like ) Morphism of G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) holds G : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) * F : ( ( ) ( LModMorphism-like ) Morphism of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) is ( ( strict ) ( strict LModMorphism-like ) Morphism of G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ;

definition
let R be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ;
let G1, G2, G3 be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;
let G be ( ( ) ( LModMorphism-like ) Morphism of G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ;
let F be ( ( ) ( LModMorphism-like ) Morphism of G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ;
func G *' F -> ( ( strict ) ( strict LModMorphism-like ) Morphism of G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,G3 : ( ( V6() quasi_total ) ( V1() V4([:G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) :] : ( ( ) ( ) set ) ) V5(G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) V6() quasi_total ) Element of bool [:[:G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) :] : ( ( ) ( ) set ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) equals :: MOD_2:def 11
G : ( ( ) ( ) Element of G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) * F : ( ( V6() quasi_total ) ( V1() V4([: the carrier of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) :] : ( ( ) ( ) set ) ) V5(G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ) V6() quasi_total ) Element of bool [:[: the carrier of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) :] : ( ( ) ( ) set ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) ;
end;

theorem :: MOD_2:13
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G2, G3, G1 being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for G being ( ( ) ( LModMorphism-like ) Morphism of G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) )
for F being ( ( ) ( LModMorphism-like ) Morphism of G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) )
for g being ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( V6() quasi_total ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st G : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) = LModMorphismStr(# G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,g : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & F : ( ( ) ( LModMorphism-like ) Morphism of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) = LModMorphismStr(# G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
( G : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) *' F : ( ( ) ( LModMorphism-like ) Morphism of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict ) ( strict LModMorphism-like ) Morphism of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) = LModMorphismStr(# G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,(g : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( V6() ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & G : ( ( ) ( LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) * F : ( ( ) ( LModMorphism-like ) Morphism of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = LModMorphismStr(# G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,(g : ( ( V6() quasi_total ) ( V1() V4( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( V6() quasi_total ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( V6() ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ;

theorem :: MOD_2:14
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for f, g being ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) st dom g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = cod f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
ex G1, G2, G3 being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ex f0 being ( ( V6() quasi_total ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ex g0 being ( ( V6() quasi_total ) ( V1() V4( the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b6 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st
( f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = LModMorphismStr(# G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,f0 : ( ( V6() quasi_total ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = LModMorphismStr(# G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,g0 : ( ( V6() quasi_total ) ( V1() V4( the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b6 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) * f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = LModMorphismStr(# G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,(g0 : ( ( V6() quasi_total ) ( V1() V4( the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b6 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f0 : ( ( V6() quasi_total ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( V6() ) ( V1() V4( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of b6 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) , the carrier of b6 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LModMorphismStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ;

theorem :: MOD_2:15
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for f, g being ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) st dom g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = cod f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
( dom (g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) * f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = dom f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & cod (g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) * f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = cod g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ;

theorem :: MOD_2:16
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G1, G2, G3, G4 being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) )
for f being ( ( strict ) ( strict LModMorphism-like ) Morphism of G1 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) )
for g being ( ( strict ) ( strict LModMorphism-like ) Morphism of G2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) )
for h being ( ( strict ) ( strict LModMorphism-like ) Morphism of G3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,G4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) holds h : ( ( strict ) ( strict LModMorphism-like ) Morphism of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) * (g : ( ( strict ) ( strict LModMorphism-like ) Morphism of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) * f : ( ( strict ) ( strict LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = (h : ( ( strict ) ( strict LModMorphism-like ) Morphism of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) * g : ( ( strict ) ( strict LModMorphism-like ) Morphism of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) * f : ( ( strict ) ( strict LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;

theorem :: MOD_2:17
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for f, g, h being ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) st dom h : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = cod g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & dom g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = cod f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
h : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) * (g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) * f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = (h : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) * g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) * f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ;

theorem :: MOD_2:18
for R being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring)
for G being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
( dom (ID G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict ) ( strict LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & cod (ID G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict ) ( strict LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) & ( for f being ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) st cod f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
(ID G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict ) ( strict LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) * f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = f : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) & ( for g being ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of R : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) st dom g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) holds
g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) * (ID G : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict ) ( strict LModMorphism-like ) Morphism of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ,b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) = g : ( ( strict LModMorphism-like ) ( strict LModMorphism-like ) LModMorphism of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Ring) ) ) ) ;

theorem :: MOD_2:19
for UN being ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe)
for u, v, w being ( ( ) ( ) Element of UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) holds {u : ( ( ) ( ) Element of b1 : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) ,v : ( ( ) ( ) Element of b1 : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) ,w : ( ( ) ( ) Element of b1 : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) } : ( ( ) ( non empty ) set ) is ( ( ) ( ) Element of UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) ;

theorem :: MOD_2:20
for UN being ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe)
for u being ( ( ) ( ) Element of UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) holds succ u : ( ( ) ( ) Element of b1 : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) : ( ( ) ( ) set ) is ( ( ) ( ) Element of UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) ;

theorem :: MOD_2:21
for UN being ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) holds
( 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) is ( ( ) ( ) Element of UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) & 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) is ( ( ) ( ) Element of UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) & 2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) is ( ( ) ( ) Element of UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) ) ;

definition
let a be ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ;
func - a -> ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) equals :: MOD_2:def 12
 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )
2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )
1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )
;
let b be ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ;
func a + b -> ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) equals :: MOD_2:def 13
b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) if a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )
a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) if b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )
2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if ( a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) )
0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if ( a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = 2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) )
0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if ( a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) )
1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if ( a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = 2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) )
;
func a * b -> ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) equals :: MOD_2:def 14
 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )
0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )
a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) if b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )
b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) if a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )
1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) if ( a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) = 2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) VectSpStr over a : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) = 2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) )
;
end;

definition
func add3 -> ( ( V6() quasi_total ) ( V1() V4([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5({0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) V6() non empty V14([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) BinOp of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) means :: MOD_2:def 15
 for a, b being ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) holds it : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) . (a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + b : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ;
func mult3 -> ( ( V6() quasi_total ) ( V1() V4([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5({0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) V6() non empty V14([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) BinOp of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) means :: MOD_2:def 16
 for a, b being ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) holds it : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) . (a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * b : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ;
func compl3 -> ( ( V6() quasi_total ) ( V1() V4({0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) V5({0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) V6() non empty V14({0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) quasi_total ) UnOp of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) means :: MOD_2:def 17
 for a being ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) holds it : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V89() associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) . a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = - a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ;
func unit3 -> ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) equals :: MOD_2:def 18
1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ;
func zero3 -> ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) equals :: MOD_2:def 19
 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
func Z_3 -> ( ( strict ) ( strict ) doubleLoopStr ) equals :: MOD_2:def 20
 doubleLoopStr(# {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,add3 : ( ( V6() quasi_total ) ( V1() V4([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5({0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) V6() non empty V14([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) BinOp of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ,mult3 : ( ( V6() quasi_total ) ( V1() V4([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5({0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) V6() non empty V14([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) BinOp of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ,unit3 : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ,zero3 : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ;
end;

registration
cluster Z_3 : ( ( strict ) ( strict ) doubleLoopStr ) -> non empty strict ;
end;

registration
cluster Z_3 : ( ( strict ) ( non empty strict ) doubleLoopStr ) -> strict well-unital ;
end;

theorem :: MOD_2:22
( 0. Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( V46( Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) ) ) Element of the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) & 1. Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( ) Element of the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) & 0. Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( V46( Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) ) ) Element of the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & 1. Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( ) Element of the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & the addF of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( V6() quasi_total ) ( V1() V4([: the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5( the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) V6() non empty V14([: the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = add3 : ( ( V6() quasi_total ) ( V1() V4([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5({0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) V6() non empty V14([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) BinOp of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & the multF of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( V6() quasi_total ) ( V1() V4([: the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5( the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) V6() non empty V14([: the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = mult3 : ( ( V6() quasi_total ) ( V1() V4([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5({0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) V6() non empty V14([:{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,{0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) BinOp of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) ;

registration
cluster Z_3 : ( ( strict ) ( non empty strict V89() right_unital well-unital left_unital ) doubleLoopStr ) -> right_complementable strict add-associative right_zeroed ;
end;

theorem :: MOD_2:23
for x, y being ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) )
for X, Y being ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) st X : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) & Y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = y : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) holds
( x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = X : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + Y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = X : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * Y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & - x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = - X : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MOD_2:24
for x, y, z being ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) )
for X, Y, Z being ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) st X : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) & Y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = y : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) & Z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = z : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) holds
( (x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) + z : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (X : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + Y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + Z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + (y : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + z : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = X : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + (Y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + Z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & (x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * z : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (X : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * Y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * Z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * (y : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * z : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = X : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * (Y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * Z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MOD_2:25
for x, y, z, a, b being ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) & b : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = 1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & (x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + (y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + (- x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & (x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * (y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & ( x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) <> a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) implies ex y being ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = b : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) & a : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) <> b : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * (y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) = (x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) + (x : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) * z : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of {0 : ( ( ) ( empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ,2 : ( ( ) ( non empty ) Element of K100() : ( ( ) ( ) Element of bool K96() : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MOD_2:26
for F being ( ( non empty ) ( non empty ) doubleLoopStr ) st ( for x, y, z being ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) holds
( x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & (x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) + z : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + (y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + z : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + (0. F : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( V46(b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + (- x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = 0. F : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( V46(b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & (x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * z : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * (y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * z : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & (1. F : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * (1. F : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) & ( x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) <> 0. F : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( V46(b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) implies ex y being ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = 1. F : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) & 0. F : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( V46(b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) <> 1. F : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * (y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + z : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) + (x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * z : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ) holds
F : ( ( non empty ) ( non empty ) doubleLoopStr ) is ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V162() V163() V164() V165() ) Field) ;

theorem :: MOD_2:27
Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) is ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Fanoian Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) Field) ;

registration
cluster Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V89() right_unital well-unital left_unital add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) -> right_complementable almost_left_invertible strict associative commutative well-unital distributive Fanoian Abelian add-associative right_zeroed ;
end;

theorem :: MOD_2:28
for D, D9 being ( ( non empty ) ( non empty ) set )
for UN being ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe)
for f being ( ( V6() quasi_total ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5(b2 : ( ( non empty ) ( non empty ) set ) ) V6() non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) quasi_total ) Function of D : ( ( non empty ) ( non empty ) set ) ,D9 : ( ( non empty ) ( non empty ) set ) ) st D : ( ( non empty ) ( non empty ) set ) in UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) & D9 : ( ( non empty ) ( non empty ) set ) in UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) holds
f : ( ( V6() quasi_total ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5(b2 : ( ( non empty ) ( non empty ) set ) ) V6() non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) quasi_total ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) in UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ;

theorem :: MOD_2:29
for UN being ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) holds
( the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) in UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) & the addF of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( V6() quasi_total ) ( V1() V4([: the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5( the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) V6() non empty V14([: the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) & comp Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( V6() quasi_total ) ( V1() V4( the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) V5( the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) V6() non empty V14( the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) & 0. Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( V46( Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) ) left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) in UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) & the multF of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( V6() quasi_total ) ( V1() V4([: the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) V5( the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) V6() non empty V14([: the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) & 1. Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of the carrier of Z_3 : ( ( strict ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible strict V89() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian Abelian add-associative right_zeroed V162() V163() V164() V165() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) in UN : ( ( non empty universal ) ( non empty epsilon-transitive universal V169() V170() ) Universe) ) ;