:: MOD_4 semantic presentation

begin

registration
let G be ( ( non empty ) ( non empty ) addMagma ) ;
cluster id G : ( ( non empty ) ( non empty ) addMagma ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of G : ( ( non empty ) ( non empty ) addMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of G : ( ( non empty ) ( non empty ) addMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of G : ( ( non empty ) ( non empty ) addMagma ) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Element of bool [: the carrier of G : ( ( non empty ) ( non empty ) addMagma ) : ( ( ) ( non empty ) set ) , the carrier of G : ( ( non empty ) ( non empty ) addMagma ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -> Function-like quasi_total bijective ;
end;

definition
let A, B, C be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like quasi_total ) ( non empty Relation-like [:A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V17([:A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [:A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ;
:: original: ~
redefine func ~ f -> ( ( Function-like quasi_total ) ( non empty Relation-like [:B : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined C : ( ( ) ( ) M29(A : ( ( ) ( ) set ) ,B : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like V17([:B : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [:B : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,C : ( ( ) ( ) M29(A : ( ( ) ( ) set ) ,B : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) ;
end;

theorem :: MOD_4:1
for C, A, B being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like quasi_total ) ( non empty Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V17([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [:A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,C : ( ( non empty ) ( non empty ) set ) )
for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) )
for y being ( ( ) ( ) Element of B : ( ( non empty ) ( non empty ) set ) ) holds f : ( ( Function-like quasi_total ) ( non empty Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V17([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) = (~ f : ( ( Function-like quasi_total ) ( non empty Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V17([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V17([:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (y : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) ,x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ;

begin

definition
let K be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
func opp K -> ( ( strict ) ( strict ) doubleLoopStr ) equals :: MOD_4:def 1
 doubleLoopStr(# the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the addF of K : ( ( ) ( ) set ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,(~ the multF of K : ( ( ) ( ) set ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V17([: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ,(1. K : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ,(0. K : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) #) : ( ( strict ) ( strict ) doubleLoopStr ) ;
end;

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

registration
let K be ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) doubleLoopStr ) ;
cluster opp K : ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) -> strict well-unital ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ;
cluster opp K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) -> right_complementable strict add-associative right_zeroed ;
end;

theorem :: MOD_4:2
for K being ( ( non empty ) ( non empty ) doubleLoopStr ) holds
( addLoopStr(# the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the addF of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) addLoopStr ) = addLoopStr(# the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the addF of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) addLoopStr ) & ( K : ( ( non empty ) ( non empty ) doubleLoopStr ) is add-associative & K : ( ( non empty ) ( non empty ) doubleLoopStr ) is right_zeroed & K : ( ( non empty ) ( non empty ) doubleLoopStr ) is right_complementable implies comp (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = comp K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) is ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) iff x : ( ( ) ( ) set ) is ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: MOD_4:3
( ( for K being ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) holds 1. K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = 1. (opp K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) & ( for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) holds
( 0. K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( V46(b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = 0. (opp K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( V46( opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) ) ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & ( for x, y, z, u being ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) )
for a, b, c, d being ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) & y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) = b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) & z : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) = c : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) & u : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) = d : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) holds
( x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & - x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = - a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & (x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) + z : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) + c : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + (y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + z : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + (b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + c : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & (x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * z : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = c : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * (b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * (y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * z : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (c : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * (y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + z : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + c : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & (y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + z : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * (b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) + c : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & (x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) + (z : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * u : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (b : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * a : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) + (d : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * c : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (opp b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ) ) ) ) ;

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

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

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

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

registration
let K be ( ( non empty associative ) ( non empty associative ) doubleLoopStr ) ;
cluster opp K : ( ( non empty associative ) ( non empty associative ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) -> strict associative ;
end;

registration
let K be ( ( non empty distributive ) ( non empty right-distributive left-distributive distributive ) doubleLoopStr ) ;
cluster opp K : ( ( non empty distributive ) ( non empty right-distributive left-distributive distributive ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) -> strict distributive ;
end;

theorem :: MOD_4:4
for K being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) holds opp K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) is ( ( non empty right_complementable strict associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ;

theorem :: MOD_4:5
for K being ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) holds opp K : ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) is ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) ;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) ;
cluster opp K : ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) -> non degenerated right_complementable almost_left_invertible strict unital associative distributive Abelian add-associative right_zeroed ;
end;

theorem :: MOD_4:6
for K being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) holds opp K : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) : ( ( strict ) ( non empty non degenerated non trivial right_complementable almost_left_invertible strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) is ( ( non empty non degenerated right_complementable almost_left_invertible strict associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ;
cluster opp K : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( strict ) ( non empty non degenerated non trivial right_complementable almost_left_invertible strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) -> almost_left_invertible strict ;
end;

begin

definition
let K be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ;
func opp V -> ( ( strict ) ( strict ) RightModStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) means :: MOD_4:def 2
 for o being ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like V17([: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) st o : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = ~ the lmult of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like V17([: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) holds
it : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) = RightModStr(# the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the addF of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,(0. V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( ) ( ) Element of the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) ,o : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) RightModStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) ;
end;

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

theorem :: MOD_4:7
for K being ( ( non empty ) ( non empty ) doubleLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) holds
( addLoopStr(# the carrier of (opp V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the addF of (opp V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of (opp V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) addLoopStr ) = addLoopStr(# the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the addF of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) addLoopStr ) & ( for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) is ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) iff x : ( ( ) ( ) set ) is ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) ) ) ;

definition
let K be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ;
let o be ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;
func opp o -> ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) RightModStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) , the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of (opp V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) RightModStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) -valued Function-like V17([: the carrier of (opp V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) RightModStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) , the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [: the carrier of (opp V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) RightModStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) , the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of (opp V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) RightModStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) ) equals :: MOD_4:def 3
 ~ o : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like V17([: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [: the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) ;
end;

theorem :: MOD_4:8
for K being ( ( non empty ) ( non empty ) doubleLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) holds the rmult of (opp V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = opp the lmult of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

definition
let K be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let W be ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ;
func opp W -> ( ( strict ) ( strict ) VectSpStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) means :: MOD_4:def 4
 for o being ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like V17([: the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [: the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) st o : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = ~ the rmult of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like V17([: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) holds
it : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) = VectSpStr(# the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the addF of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,(0. W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( ) ( ) Element of the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) ,o : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of (opp K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) VectSpStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) ;
end;

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

theorem :: MOD_4:9
for K being ( ( non empty ) ( non empty ) doubleLoopStr )
for W being ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) holds
( addLoopStr(# the carrier of (opp W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the addF of (opp W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of (opp W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) addLoopStr ) = addLoopStr(# the carrier of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the addF of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) addLoopStr ) & ( for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) is ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) iff x : ( ( ) ( ) set ) is ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) ) ) ;

definition
let K be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let W be ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ;
let o be ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;
func opp o -> ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) , the carrier of (opp W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) VectSpStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of (opp W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) VectSpStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) -valued Function-like V17([: the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) , the carrier of (opp W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) VectSpStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [: the carrier of (opp K : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) , the carrier of (opp W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) VectSpStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of (opp W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) VectSpStr over opp K : ( ( ) ( ) set ) : ( ( strict ) ( strict ) doubleLoopStr ) ) : ( ( ) ( ) set ) ) equals :: MOD_4:def 5
 ~ o : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like V17([: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) quasi_total ) Function of [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of W : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) ;
end;

theorem :: MOD_4:10
for K being ( ( non empty ) ( non empty ) doubleLoopStr )
for W being ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) ) holds the lmult of (opp W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = opp the rmult of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MOD_4:11
for K being ( ( non empty ) ( non empty ) doubleLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) )
for o being ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) )
for y being ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) )
for w being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) & v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) = w : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds
(opp o : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) . (w : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = o : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) . (x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MOD_4:12
for K, L being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring)
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for W being ( ( non empty ) ( non empty ) RightModStr over L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for x being ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) )
for y being ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) )
for w being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) st L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) = opp K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) & W : ( ( non empty ) ( non empty ) RightModStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) = opp V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) & x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) & v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) = w : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds
w : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b4 : ( ( non empty ) ( non empty ) RightModStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of 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 right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MOD_4:13
for K, L being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring)
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for W being ( ( non empty ) ( non empty ) RightModStr over L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for v1, v2 being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) )
for w1, w2 being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) st W : ( ( non empty ) ( non empty ) RightModStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) = opp V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) & v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) = w1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) & v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) = w2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds
w1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) + w2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b4 : ( ( non empty ) ( non empty ) RightModStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) = v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) + v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of 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 right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MOD_4:14
for K being ( ( non empty ) ( non empty ) doubleLoopStr )
for W being ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty ) ( non empty ) doubleLoopStr ) )
for o being ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) )
for y being ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) )
for w being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) & v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) = w : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds
(opp o : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of (opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) . (y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ,w : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (opp b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = o : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Function of [: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) . (v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ,x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MOD_4:15
for K, L being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring)
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for W being ( ( non empty ) ( non empty ) RightModStr over L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for x being ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) )
for y being ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) )
for w being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) st V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) = opp W : ( ( non empty ) ( non empty ) RightModStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) & x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) & v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) = w : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds
w : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b4 : ( ( non empty ) ( non empty ) RightModStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) = x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of 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 right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MOD_4:16
for K, L being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring)
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for W being ( ( non empty ) ( non empty ) RightModStr over L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for v1, v2 being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) )
for w1, w2 being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) st V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) = opp W : ( ( non empty ) ( non empty ) RightModStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) & v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) = w1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) & v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) = w2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds
w1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) + w2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b4 : ( ( non empty ) ( non empty ) RightModStr over b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) = v1 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) + v2 : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of 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 right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MOD_4:17
for K being ( ( non empty strict ) ( non empty strict ) doubleLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) holds opp (opp V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp (opp b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) = VectSpStr(# the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the addF of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) , the lmult of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) VectSpStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) ;

theorem :: MOD_4:18
for K being ( ( non empty strict ) ( non empty strict ) doubleLoopStr )
for W being ( ( non empty ) ( non empty ) RightModStr over K : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) holds opp (opp W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp (opp b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) : ( ( strict ) ( non empty strict ) doubleLoopStr ) ) = RightModStr(# the carrier of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the addF of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the ZeroF of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) , the rmult of W : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like V17([: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [:[: the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) RightModStr over b1 : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) ;

theorem :: MOD_4:19
errorfrm ;

registration
let K be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ;
cluster opp V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) VectSpStr over K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) RightModStr over opp K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) -> right_complementable Abelian add-associative right_zeroed strict RightMod-like ;
end;

theorem :: MOD_4:20
for K being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring)
for W being ( ( non empty right_complementable Abelian add-associative right_zeroed RightMod-like ) ( non empty right_complementable Abelian add-associative right_zeroed RightMod-like ) RightMod of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) holds opp W : ( ( non empty right_complementable Abelian add-associative right_zeroed RightMod-like ) ( non empty right_complementable Abelian add-associative right_zeroed RightMod-like ) RightMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) is ( ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of opp K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ;

registration
let K be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ;
let W be ( ( non empty right_complementable Abelian add-associative right_zeroed RightMod-like ) ( non empty right_complementable Abelian add-associative right_zeroed RightMod-like ) RightMod of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ;
cluster opp W : ( ( non empty right_complementable Abelian add-associative right_zeroed RightMod-like ) ( non empty right_complementable Abelian add-associative right_zeroed RightMod-like ) RightModStr over K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) VectSpStr over opp K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable strict unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) -> right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ;
end;

begin

definition
let K, L be ( ( non empty ) ( non empty ) multMagma ) ;
let IT be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of K : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
attr IT is antimultiplicative means :: MOD_4:def 6
 for x, y being ( ( ) ( ) Scalar of ( ( ) ( ) set ) ) holds IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) . (x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) * y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) = (IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) . y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) * (IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) . x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) ;
end;

definition
let K, L be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let IT be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
attr IT is antilinear means :: MOD_4:def 7
( IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is additive & IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is antimultiplicative & IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is unity-preserving );
end;

registration
let K, L be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
cluster Function-like quasi_total antilinear -> Function-like quasi_total unity-preserving additive antimultiplicative for ( ( ) ( ) Element of bool [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
cluster Function-like quasi_total unity-preserving additive antimultiplicative -> Function-like quasi_total antilinear for ( ( ) ( ) Element of bool [: the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let K, L be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let IT be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
attr IT is monomorphism means :: MOD_4:def 8
( IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is linear & IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is one-to-one );
attr IT is antimonomorphism means :: MOD_4:def 9
( IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is antilinear & IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is one-to-one );
end;

definition
let K, L be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let IT be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
attr IT is epimorphism means :: MOD_4:def 10
( IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is linear & IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is onto );
attr IT is antiepimorphism means :: MOD_4:def 11
( IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is antilinear & IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is onto );
end;

definition
let K, L be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let IT be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
attr IT is isomorphism means :: MOD_4:def 12
( IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is monomorphism & IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is onto );
attr IT is antiisomorphism means :: MOD_4:def 13
( IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is antimonomorphism & IT : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) is onto );
end;

definition
let K be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let IT be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
attr IT is endomorphism means :: MOD_4:def 14
IT : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) is linear ;
attr IT is antiendomorphism means :: MOD_4:def 15
IT : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) is antilinear ;
end;

theorem :: MOD_4:21
for K being ( ( non empty ) ( non empty ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is isomorphism iff ( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is additive & ( for x, y being ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) holds J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( 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 ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . y : ( ( ) ( ) 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 ) ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (1_ K : ( ( 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 ) ) = 1_ K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is onto ) ) ;

theorem :: MOD_4:22
for K being ( ( non empty ) ( non empty ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antiisomorphism iff ( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is additive & ( for x, y being ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) holds J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( 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 ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . y : ( ( ) ( ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( 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 ) ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (1_ K : ( ( 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 ) ) = 1_ K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is onto ) ) ;

registration
let K be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
cluster id K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one V17( the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total onto bijective unity-preserving additive multiplicative linear ) Element of bool [: the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -> Function-like quasi_total isomorphism ;
end;

theorem :: MOD_4:23
for K, L being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring)
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for x, y being ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is linear holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (0. K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( V46(b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) = 0. L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( V46(b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (- x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) = - (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) = (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) - (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MOD_4:24
for K, L being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring)
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for x, y being ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antilinear holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (0. K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( V46(b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) = 0. L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( V46(b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (- x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) = - (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) = (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) - (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . y : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MOD_4:25
for K being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) holds
( id K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total onto bijective unity-preserving additive multiplicative linear isomorphism ) Element of bool [: the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is antiisomorphism iff K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) is ( ( non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) comRing) ) ;

theorem :: MOD_4:26
for K being ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) holds
( id K : ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like one-to-one V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) : ( ( ) ( non empty non trivial ) set ) ) quasi_total onto bijective unity-preserving additive multiplicative linear isomorphism ) Element of bool [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is antiisomorphism iff K : ( ( non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Skew-Field) 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 right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Field) ) ;

begin

definition
let K, L be ( ( non empty ) ( non empty ) doubleLoopStr ) ;
let J be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of K : ( ( non empty ) ( non empty ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
func opp J -> ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined the carrier of (opp L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) : ( ( strict ) ( strict ) doubleLoopStr ) : ( ( ) ( ) set ) -valued Function-like V17( the carrier of K : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) quasi_total ) Function of ( ( ) ( ) set ) , ( ( ) ( ) set ) ) equals :: MOD_4:def 16
J : ( ( ) ( ) M29(K : ( ( ) ( ) set ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ;
end;

theorem :: MOD_4:27
canceled;

theorem :: MOD_4:28
for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr )
for L being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is linear iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antilinear ) ;

theorem :: MOD_4:29
for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr )
for L being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antilinear iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is linear ) ;

theorem :: MOD_4:30
for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr )
for L being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is monomorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antimonomorphism ) ;

theorem :: MOD_4:31
for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr )
for L being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antimonomorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is monomorphism ) ;

theorem :: MOD_4:32
for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr )
for L being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is epimorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antiepimorphism ) ;

theorem :: MOD_4:33
for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr )
for L being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antiepimorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is epimorphism ) ;

theorem :: MOD_4:34
for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr )
for L being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is isomorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antiisomorphism ) ;

theorem :: MOD_4:35
for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr )
for L being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antiisomorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b2 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is isomorphism ) ;

theorem :: MOD_4:36
for K being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is endomorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antilinear ) ;

theorem :: MOD_4:37
for K being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antiendomorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is linear ) ;

theorem :: MOD_4:38
for K being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is isomorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antiisomorphism ) ;

theorem :: MOD_4:39
for K being ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antiisomorphism iff opp J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (opp b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable strict unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable well-unital add-associative right_zeroed ) ( non empty right_complementable unital right_unital well-unital left_unital add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is isomorphism ) ;

begin

definition
let G, H be ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ;
mode Homomorphism of G,H is ( ( Function-like quasi_total additive ) ( non empty Relation-like the carrier of G : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined the carrier of H : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like V17( the carrier of G : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) quasi_total additive ) Function of ( ( ) ( ) set ) , ( ( ) ( ) set ) ) ;
end;

definition
let G be ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ;
mode Endomorphism of G is ( ( Function-like quasi_total additive ) ( non empty Relation-like the carrier of G : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined the carrier of G : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V17( the carrier of G : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) quasi_total additive ) Homomorphism of G : ( ( ) ( ) set ) ,G : ( ( ) ( ) set ) ) ;
end;

registration
let G be ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ;
cluster non empty Relation-like the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total bijective additive for ( ( ) ( ) Element of bool [: the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let G be ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ;
mode Automorphism of G is ( ( Function-like quasi_total bijective additive ) ( non empty Relation-like the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one V17( the carrier of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total onto bijective additive ) Endomorphism of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) ;
end;

theorem :: MOD_4:40
for G, H being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup)
for f being ( ( Function-like quasi_total additive ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Homomorphism of G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ,H : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) )
for x, y being ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like quasi_total additive ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Homomorphism of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ,b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ) . (0. G : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ) : ( ( ) ( V46(b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) = 0. H : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( V46(b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ) right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total additive ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Homomorphism of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ,b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ) . (- x : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) = - (f : ( ( Function-like quasi_total additive ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Homomorphism of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ,b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ) . x : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like quasi_total additive ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Homomorphism of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ,b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ) . (x : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like quasi_total additive ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Homomorphism of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ,b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ) . x : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) - (f : ( ( Function-like quasi_total additive ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) quasi_total additive ) Homomorphism of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ,b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) ) . y : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) AddGroup) : ( ( ) ( non empty ) set ) ) ) ;

begin

definition
let K, L be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ;
let J be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ;
let W be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ;
mode Homomorphism of J,V,W -> ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of V : ( ( Function-like quasi_total ) ( Relation-like [:J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) -defined J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like quasi_total ) Element of bool [:[:J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -defined the carrier of W : ( ( ) ( ) Element of J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) : ( ( ) ( ) set ) -valued Function-like V17( the carrier of V : ( ( Function-like quasi_total ) ( Relation-like [:J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) -defined J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like quasi_total ) Element of bool [:[:J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) quasi_total ) Function of ( ( ) ( ) set ) , ( ( ) ( ) set ) ) means :: MOD_4:def 17
( ( for x, y being ( ( ) ( right_complementable ) Vector of ( ( ) ( ) set ) ) holds it : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) -defined J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . (x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( Function-like quasi_total ) ( Relation-like [:J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) -defined J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like quasi_total ) Element of bool [:[:J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of W : ( ( ) ( ) Element of J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) : ( ( ) ( ) set ) ) = (it : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) -defined J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of W : ( ( ) ( ) Element of J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) : ( ( ) ( ) set ) ) + (it : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) -defined J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of W : ( ( ) ( ) Element of J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of W : ( ( ) ( ) Element of J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) : ( ( ) ( ) set ) ) ) & ( for a being ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( right_complementable ) Vector of ( ( ) ( ) set ) ) holds it : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) -defined J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . (a : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * x : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( Function-like quasi_total ) ( Relation-like [:J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) -defined J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like quasi_total ) Element of bool [:[:J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of W : ( ( ) ( ) Element of J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) : ( ( ) ( ) set ) ) = (J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) . a : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) * (it : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) -defined J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) -valued Function-like quasi_total ) Element of bool [:[: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) ,J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of W : ( ( ) ( ) Element of J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of W : ( ( ) ( ) Element of J : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,L : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) : ( ( ) ( ) set ) ) ) );
end;

theorem :: MOD_4:41
for K, L being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring)
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for V being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for W being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of L : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) holds ZeroMap (V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ,W : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total additive ) 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 right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) , the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( non empty Relation-like the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b5 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b4 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Homomorphism of J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ,V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ,W : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b2 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ) ;

definition
let K be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ;
let J be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -defined the carrier of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ;
mode Endomorphism of J,V is ( ( ) ( non empty Relation-like the carrier of V : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,J : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) : ( ( ) ( ) set ) -defined the carrier of V : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,J : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) : ( ( ) ( ) set ) -valued Function-like V17( the carrier of V : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,J : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) : ( ( ) ( ) set ) ) quasi_total ) Homomorphism of J : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ,V : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,J : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ,V : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,J : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) ;
end;

definition
let K be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ;
let V, W be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ;
mode Homomorphism of V,W is ( ( ) ( non empty Relation-like the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -defined the carrier of W : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) : ( ( ) ( ) set ) -valued Function-like V17( the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) quasi_total ) Homomorphism of id K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one V17( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) quasi_total onto bijective additive ) Element of bool [: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ,W : ( ( ) ( ) M29(K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ,V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) )) ) ;
end;

theorem :: MOD_4:42
for K being ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring)
for V, W being ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) )
for f being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( ) ( non empty Relation-like the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Homomorphism of V : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ,W : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ) iff ( ( for x, y being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) + (f : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . y : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) ) & ( for a being ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (a : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * x : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( right_complementable ) Scalar of ( ( ) ( non empty ) set ) ) * (f : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) -valued Function-like V17( the carrier of b2 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( right_complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of b1 : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) : ( ( ) ( non empty ) set ) ) ) ) ) ;

definition
let K be ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ;
let V be ( ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) ( non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ) LeftMod of K : ( ( non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed ) ( non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed ) Ring) ) ;
mode Endomorphism of V is ( ( ) ( non empty Relation-like the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -defined the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -valued Function-like V17( the carrier of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) quasi_total ) Homomorphism of V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ,V : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) ;
end;