:: PARSP_1 semantic presentation

begin

definition
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
func c3add F -> ( ( V6() V18([:[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ) ( V6() V18([:[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ) BinOp of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) means :: PARSP_1:def 1
 for x, y being ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) holds it : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) . (x : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) = [((x : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) + (y : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ,((x : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) + (y : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ,((x : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) + (y : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ;
end;

definition
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
let x, y be ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ;
func x + y -> ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) equals :: PARSP_1:def 2
(c3add F : ( ( ) ( ) 1-sorted ) ) : ( ( V6() V18([:[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ) ( V6() V18([:[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ) BinOp of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) . (x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) ,y : ( ( V6() V18([:x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) ,x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) ) ) ( V6() V18([:x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) ,x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) ,x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ;
end;

theorem :: PARSP_1:1
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for x, y being ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) holds x : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) = [((x : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (y : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,((x : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (y : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,((x : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (y : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:2
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, c, f, g, h being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) holds [a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) + [f : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ,g : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ,h : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) = [(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) + f : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,(b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) + g : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,(c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) + h : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ;

definition
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
func c3compl F -> ( ( V6() V18([: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ) ( V6() V18([: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ) UnOp of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) means :: PARSP_1:def 3
 for x being ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) holds it : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) . x : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) = [(- (x : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ,(- (x : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ,(- (x : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ;
end;

definition
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
let x be ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ;
func - x -> ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) equals :: PARSP_1:def 4
(c3compl F : ( ( ) ( ) 1-sorted ) ) : ( ( V6() V18([: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ) ( V6() V18([: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ) UnOp of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) . x : ( ( ) ( ) VectSpStr over F : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) Element of [: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) ;
end;

theorem :: PARSP_1:3
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for x being ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) holds - x : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) = [(- (x : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,(- (x : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,(- (x : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ;

definition
let S be ( ( ) ( ) set ) ;
mode Relation4 of S -> ( ( ) ( ) set ) means :: PARSP_1:def 5
it : ( ( ) ( ) VectSpStr over S : ( ( ) ( ) 1-sorted ) ) c= [:S : ( ( ) ( ) 1-sorted ) ,S : ( ( ) ( ) 1-sorted ) ,S : ( ( ) ( ) 1-sorted ) ,S : ( ( ) ( ) 1-sorted ) :] : ( ( ) ( ) set ) ;
end;

definition
attr c1 is strict ;
struct ParStr -> ( ( ) ( ) 1-sorted ) ;
aggr ParStr(# carrier, 4_arg_relation #) -> ( ( strict ) ( strict ) ParStr ) ;
sel 4_arg_relation c1 -> ( ( ) ( ) Relation4 of the carrier of c1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

registration
cluster non empty for ( ( ) ( ) ParStr ) ;
end;

definition
let PS be ( ( non empty ) ( non empty ) ParStr ) ;
let a, b, c, d be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
pred a,b '||' c,d means :: PARSP_1:def 6
[a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) ,b : ( ( V6() V18([:a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) ) ) ( V6() V18([:a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) ) ,d : ( ( V6() V18([: the carrier of PS : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) ) ) ( V6() V18([: the carrier of PS : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[: the carrier of PS : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,a : ( ( ) ( ) VectSpStr over PS : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of PS : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of PS : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of PS : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of PS : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) in the 4_arg_relation of PS : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) Relation4 of the carrier of PS : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

definition
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
func C_3 F -> ( ( ) ( ) set ) equals :: PARSP_1:def 7
[: the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of F : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ;
end;

registration
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
cluster C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( ) set ) -> non empty ;
end;

definition
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
func 4C_3 F -> ( ( ) ( ) set ) equals :: PARSP_1:def 8
[:(C_3 F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( ) set ) ,(C_3 F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( ) set ) ,(C_3 F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( ) set ) ,(C_3 F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ;
end;

registration
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
cluster 4C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( ) set ) -> non empty ;
end;

definition
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
func PRs F -> ( ( ) ( ) set ) means :: PARSP_1:def 9
 for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in it : ( ( ) ( ) VectSpStr over F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) iff ( x : ( ( ) ( ) set ) in 4C_3 F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( ) set ) & ex a, b, c, d being ( ( ) ( ) Element of [: the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) st
( x : ( ( ) ( ) set ) = [a : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [:[: the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) & (((a : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (b : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * ((c : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (d : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (((c : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (d : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * ((a : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (b : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = 0. F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( V49(F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & (((a : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (b : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * ((c : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (d : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (((c : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (d : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * ((a : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (b : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = 0. F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( V49(F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) & (((a : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (b : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * ((c : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (d : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (((c : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (d : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * ((a : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) - (b : ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = 0. F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( V49(F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) ) Element of the carrier of F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ) );
end;

theorem :: PARSP_1:4
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) holds PRs F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( ) set ) c= [:(C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty ) set ) ,(C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty ) set ) ,(C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty ) set ) ,(C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ;

definition
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
func PR F -> ( ( ) ( ) Relation4 of C_3 F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( ) set ) ) equals :: PARSP_1:def 10
 PRs F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( ) set ) ;
end;

definition
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
func MPS F -> ( ( ) ( ) ParStr ) equals :: PARSP_1:def 11
 ParStr(# (C_3 F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( ) set ) ,(PR F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) ) : ( ( ) ( ) Relation4 of C_3 F : ( ( non empty right_complementable associative well-unital V102() Abelian add-associative right_zeroed ) ( non empty right_complementable V92() associative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( ) set ) ) #) : ( ( strict ) ( strict ) ParStr ) ;
end;

registration
let F be ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ;
cluster MPS F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) doubleLoopStr ) : ( ( ) ( ) ParStr ) -> non empty strict ;
end;

theorem :: PARSP_1:5
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) holds the carrier of (MPS F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) = C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty ) set ) ;

theorem :: PARSP_1:6
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) holds the 4_arg_relation of (MPS F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( ) Relation4 of the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) ) = PR F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( ) Relation4 of C_3 b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:7
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) iff [a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) in PR F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( ) Relation4 of C_3 b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: PARSP_1:8
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( [a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) in PR F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( ) Relation4 of C_3 b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty ) set ) ) iff ( [a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) in 4C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty ) set ) & ex e, f, g, h being ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) st
( [a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) = [e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [:[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) & (((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) & (((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) & (((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ) ;

theorem :: PARSP_1:9
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) iff ( [a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) in 4C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty ) set ) & ex e, f, g, h being ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) st
( [a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) = [e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [:[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) & (((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) & (((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) & (((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ) ;

theorem :: PARSP_1:10
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) holds the carrier of (MPS F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) = [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ;

theorem :: PARSP_1:11
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds [a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) in 4C_3 F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty ) set ) ;

theorem :: PARSP_1:12
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) iff ex e, f, g, h being ( ( ) ( ) Element of [: the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) st
( [a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [: the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) , the carrier of (MPS b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) : ( ( ) ( non empty strict ) ParStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) = [e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ,h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( ) Element of [:[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ,[: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ) & (((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) & (((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `1_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) & (((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (((g : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (h : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `2_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) * ((e : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (f : ( ( ) ( ) Element of [: the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( non empty ) set ) ) `3_3) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. F : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: PARSP_1:13
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:14
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:15
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, p, q, r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:16
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:17
for F being ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital V102() Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V92() associative commutative right-distributive left-distributive right_unital well-unital V102() left_unital Abelian add-associative right_zeroed ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

definition
let IT be ( ( non empty ) ( non empty ) ParStr ) ;
attr IT is ParSp-like means :: PARSP_1:def 12
 for a, b, c, d, p, q, r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & ( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) implies a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) & ( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) implies b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) & ex x being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) );
end;

registration
cluster non empty strict ParSp-like for ( ( ) ( ) ParStr ) ;
end;

definition
mode ParSp is ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParStr ) ;
end;

theorem :: PARSP_1:18
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:19
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:20
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:21
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:22
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:23
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: PARSP_1:24
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: PARSP_1:25
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) or c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) or ( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) or ( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ) holds
a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:26
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, p, q, r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:27
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: PARSP_1:28
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: PARSP_1:29
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: PARSP_1:30
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, d, p, q, r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
not p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:31
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, p, q, r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
not p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:32
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, p, r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:33
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for p, q, r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:34
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, p, q, r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:35
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, d being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:36
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp) st ( for a, b being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) holds
for p, q, r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:37
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp) st ex a, b being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & ( for c being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ) holds
for p, q, r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: PARSP_1:38
for PS being ( ( non empty ParSp-like ) ( non empty ParSp-like ) ParSp)
for a, b, c, p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
not p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) '||' p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;