:: HAHNBAN1 semantic presentation

begin

theorem :: HAHNBAN1:1
canceled;

theorem :: HAHNBAN1:2
for x1, y1, x2, y2 being ( ( real ) ( complex real ext-real ) number ) holds (x1 : ( ( real ) ( complex real ext-real ) number ) + (y1 : ( ( real ) ( complex real ext-real ) number ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) ) : ( ( ) ( complex ) set ) ) : ( ( ) ( complex ) set ) * (x2 : ( ( real ) ( complex real ext-real ) number ) + (y2 : ( ( real ) ( complex real ext-real ) number ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) ) : ( ( ) ( complex ) set ) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex ) set ) = ((x1 : ( ( real ) ( complex real ext-real ) number ) * x2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) - (y1 : ( ( real ) ( complex real ext-real ) number ) * y2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) set ) + (((x1 : ( ( real ) ( complex real ext-real ) number ) * y2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) + (x2 : ( ( real ) ( complex real ext-real ) number ) * y1 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) set ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex ) set ) ;

theorem :: HAHNBAN1:3
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) holds |.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) + (0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex ) set ) = ((z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) *') : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) / (|.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) + (0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) ) : ( ( ) ( complex ) set ) ) : ( ( ) ( complex ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) * z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) ;

begin

definition
let x, y be ( ( real ) ( complex real ext-real ) number ) ;
func [**x,y**] -> ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) equals :: HAHNBAN1:def 1
x : ( ( ) ( ) 1-sorted ) + (y : ( ( ) ( ) VectSpStr over x : ( ( ) ( ) 1-sorted ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;
end;

definition
func i_FC -> ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) equals :: HAHNBAN1:def 2
 <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V35() V192() V198() ) set ) ) ;
end;

theorem :: HAHNBAN1:4
i_FC : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * i_FC : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = - (1_ F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: HAHNBAN1:5
(- (1_ F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) * (- (1_ F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = 1_ F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: HAHNBAN1:6
for x1, y1, x2, y2 being ( ( ) ( complex real ext-real ) Real) holds [**x1 : ( ( ) ( complex real ext-real ) Real) ,y1 : ( ( ) ( complex real ext-real ) Real) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) + [**x2 : ( ( ) ( complex real ext-real ) Real) ,y2 : ( ( ) ( complex real ext-real ) Real) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = [**(x1 : ( ( ) ( complex real ext-real ) Real) + x2 : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ,(y1 : ( ( ) ( complex real ext-real ) Real) + y2 : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: HAHNBAN1:7
for x1, y1, x2, y2 being ( ( real ) ( complex real ext-real ) number ) holds [**x1 : ( ( real ) ( complex real ext-real ) number ) ,y1 : ( ( real ) ( complex real ext-real ) number ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * [**x2 : ( ( real ) ( complex real ext-real ) number ) ,y2 : ( ( real ) ( complex real ext-real ) number ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = [**((x1 : ( ( real ) ( complex real ext-real ) number ) * x2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) - (y1 : ( ( real ) ( complex real ext-real ) number ) * y2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) set ) ,((x1 : ( ( real ) ( complex real ext-real ) number ) * y2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) + (x2 : ( ( real ) ( complex real ext-real ) number ) * y1 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) set ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: HAHNBAN1:8
canceled;

theorem :: HAHNBAN1:9
for r being ( ( ) ( complex real ext-real ) Real) holds |.[**r : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = abs r : ( ( ) ( complex real ext-real ) Real) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;

theorem :: HAHNBAN1:10
for x, y being ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) holds
( Re (x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) + y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = (Re x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) + (Re y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) & Im (x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) + y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = (Im x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) + (Im y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ;

theorem :: HAHNBAN1:11
for x, y being ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) holds
( Re (x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = ((Re x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) * (Re y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) - ((Im x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) * (Im y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) & Im (x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = ((Re x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) * (Im y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) + ((Re y : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) * (Im x : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ;

begin

definition
let K be ( ( ) ( ) 1-sorted ) ;
let V be ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
mode Functional of V is ( ( Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( V16() V19( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) V20( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Function of the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) addLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) ;
let f, g be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) ) ;
func f + g -> ( ( Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( V16() V19( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) V20( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Functional of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) means :: HAHNBAN1:def 3
 for x being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds it : ( ( Function-like V30([: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( V16() V19([: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V20(V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) Function-like V30([: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) = (f : ( ( Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( V16() V19([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V20(V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) + (g : ( ( ) ( ) Element of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) addLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) ;
let f be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) ) ;
func - f -> ( ( Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( V16() V19( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) V20( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Functional of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) means :: HAHNBAN1:def 4
 for x being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds it : ( ( ) ( ) Element of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) = - (f : ( ( Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( V16() V19([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V20(V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) addLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) ;
let f, g be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) addLoopStr ) ) ) ;
func f - g -> ( ( Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( V16() V19( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) V20( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Functional of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) equals :: HAHNBAN1:def 5
f : ( ( Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( V16() V19([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V20(V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) + (- g : ( ( ) ( ) Element of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) : ( ( Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( V16() V19( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) V20( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Functional of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) : ( ( Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( V16() V19( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) V20( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Functional of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) multMagma ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) ;
let v be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
let f be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) ) ;
func v * f -> ( ( Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( V16() V19( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) V20( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Functional of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) means :: HAHNBAN1:def 6
 for x being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds it : ( ( Function-like V30([: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( V16() V19([: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V20(V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) Function-like V30([: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[: the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) = v : ( ( Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( V16() V19([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V20(V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) * (f : ( ( ) ( ) Element of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) ZeroStr ) ;
let V be ( ( ) ( ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ;
func 0Functional V -> ( ( Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( V16() V19( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) V20( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Functional of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) equals :: HAHNBAN1:def 7
([#] V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) : ( ( ) ( non proper ) Element of bool the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) --> (0. K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( [#] V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non proper ) Element of bool the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( V16() V19( [#] V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non proper ) Element of bool the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) V20( the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) Function-like V30( [#] V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non proper ) Element of bool the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Element of bool [:([#] V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) : ( ( ) ( non proper ) Element of bool the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) multMagma ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) ;
let F be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) multMagma ) ) ) ;
attr F is homogeneous means :: HAHNBAN1:def 8
 for x being ( ( ) ( ) Vector of ( ( ) ( ) set ) )
for r being ( ( ) ( ) Scalar of ) holds F : ( ( Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( V16() V19([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V20(V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . (r : ( ( ) ( ) Scalar of ) * x : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) = r : ( ( ) ( ) Scalar of ) * (F : ( ( Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( V16() V19([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V20(V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) ZeroStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ;
let F be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ) ;
attr F is 0-preserving means :: HAHNBAN1:def 9
F : ( ( Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( V16() V19([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V20(V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) Function-like V30([:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . (0. V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) : ( ( ) ( zero ) Element of the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) = 0. K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

registration
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() unital associative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) homogeneous -> Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) 0-preserving for ( ( ) ( ) Element of bool [: the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

registration
let K be ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ;
cluster 0Functional V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) additive ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ;
cluster 0Functional V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ;
end;

registration
let K be ( ( non empty ) ( non empty ) ZeroStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ;
cluster 0Functional V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) 0-preserving ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ;
cluster V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous 0-preserving for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

theorem :: HAHNBAN1:12
for K being ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) )
for f, g being ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) ) holds f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) ) + g : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) ) = g : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) ) + f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty Abelian ) ( non empty Abelian ) addLoopStr ) ) ) ;

theorem :: HAHNBAN1:13
for K being ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) )
for f, g, h being ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) holds (f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) + g : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) + h : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) = f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) + (g : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) + h : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty add-associative ) ( non empty add-associative ) addLoopStr ) ) ) ;

theorem :: HAHNBAN1:14
for K being ( ( non empty ) ( non empty ) ZeroStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty ) ( non empty ) ZeroStr ) )
for x being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds (0Functional V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) ZeroStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) 0-preserving ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty ) ( non empty ) ZeroStr ) ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) = 0. K : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( zero ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HAHNBAN1:15
for K being ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) )
for f being ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ) holds f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ) + (0Functional V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) additive 0-preserving ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ) = f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_zeroed ) ( non empty right_zeroed ) addLoopStr ) ) ) ;

theorem :: HAHNBAN1:16
for K being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) )
for f being ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) ) holds f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) ) - f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) ) = 0Functional V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) : ( ( ) ( non empty ) set ) ) additive 0-preserving ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) addLoopStr ) ) ) ;

theorem :: HAHNBAN1:17
for K being ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) )
for r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for f, g being ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) holds r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * (f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) + g : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) = (r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) + (r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * g : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty right-distributive ) ( non empty right-distributive ) doubleLoopStr ) ) ) ;

theorem :: HAHNBAN1:18
for K being ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) )
for r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for f being ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) ) holds (r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) + s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) ) = (r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) ) + (s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) ) ) ;

theorem :: HAHNBAN1:19
for K being ( ( non empty associative ) ( non empty associative ) multMagma )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty associative ) ( non empty associative ) multMagma ) )
for r, s being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for f being ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) ) holds (r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) ) = r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * (s : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty associative ) ( non empty associative ) multMagma ) ) ) ;

theorem :: HAHNBAN1:20
for K being ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr )
for V being ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) )
for f being ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) ) holds (1. K : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) ) : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) ) = f : ( ( Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of b2 : ( ( non empty ) ( non empty ) VectSpStr over b1 : ( ( non empty left_unital ) ( non empty left_unital ) doubleLoopStr ) ) ) ;

registration
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) ;
let f, g be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) ) ;
cluster f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) + g : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ;
end;

registration
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) ;
let f be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) ) ;
cluster - f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ;
let v be ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;
let f be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) ;
cluster v : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ;
let f, g be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) ;
cluster f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) + g : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ;
end;

registration
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) ;
let f be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) ) ;
cluster - f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ;
end;

registration
let K be ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) ;
let v be ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;
let f be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) ) ;
cluster v : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) ( non empty right_complementable add-associative right_zeroed associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) homogeneous ;
end;

definition
let K be ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ;
mode linear-Functional of V is ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) ;
end;

begin

definition
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) ;
func V *' -> ( ( non empty strict ) ( non empty strict ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) means :: HAHNBAN1:def 10
( ( for x being ( ( ) ( ) set ) holds
( x : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) linear-Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) ) in the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) iff x : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) linear-Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) ) is ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) linear-Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) ) ) & ( for f, g being ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) linear-Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) holds the addF of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V30([: the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19([: the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V20( the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) Function-like V30([: the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . (f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) linear-Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) ) ,g : ( ( ) ( left_complementable right_complementable complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) = f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) linear-Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) ) + g : ( ( ) ( left_complementable right_complementable complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) ) & 0. it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = 0Functional V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous 0-preserving ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) & ( for f being ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) linear-Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) )
for x being ( ( ) ( left_complementable right_complementable complementable ) Element of ( ( ) ( non empty ) set ) ) holds the lmult of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V30([: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19([: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ) V20( the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) Function-like V30([: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( left_complementable right_complementable complementable ) Element of ( ( ) ( non empty ) set ) ) ,f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) linear-Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) ) ) : ( ( ) ( ) set ) = x : ( ( ) ( left_complementable right_complementable complementable ) Element of ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) additive homogeneous ) linear-Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) ) );
end;

registration
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) ;
cluster V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) *' : ( ( non empty strict ) ( non empty strict ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) -> non empty Abelian strict ;
end;

registration
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) ;
cluster V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) *' : ( ( non empty strict ) ( non empty Abelian strict ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) -> non empty add-associative strict ;
cluster V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) *' : ( ( non empty strict ) ( non empty Abelian strict ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) -> non empty right_zeroed strict ;
cluster V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) *' : ( ( non empty strict ) ( non empty Abelian strict ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative right-distributive ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive ) doubleLoopStr ) ) -> non empty right_complementable strict ;
end;

registration
let K be ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative distributive left_unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital distributive left_unital ) doubleLoopStr ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative distributive left_unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital distributive left_unital ) doubleLoopStr ) ) ;
cluster V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative distributive left_unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital distributive left_unital ) doubleLoopStr ) ) *' : ( ( non empty strict ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() strict ) VectSpStr over K : ( ( non empty right_complementable Abelian add-associative right_zeroed associative commutative distributive left_unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital distributive left_unital ) doubleLoopStr ) ) -> non empty strict vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

begin

definition
let K be ( ( ) ( ) 1-sorted ) ;
let V be ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
mode RFunctional of V is ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Function of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;
end;

definition
let K be ( ( ) ( ) 1-sorted ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
let F be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ;
attr F is subadditive means :: HAHNBAN1:def 11
 for x, y being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds F : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) . (x : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) <= (F : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) + (F : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) . y : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;
end;

definition
let K be ( ( ) ( ) 1-sorted ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
let F be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ;
attr F is additive means :: HAHNBAN1:def 12
 for x, y being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds F : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) . (x : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = (F : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) + (F : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) . y : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;
end;

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let F be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
attr F is Real_homogeneous means :: HAHNBAN1:def 13
 for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( complex real ext-real ) Real) holds F : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) . ([**r : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = r : ( ( ) ( complex real ext-real ) Real) * (F : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) . v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;
end;

theorem :: HAHNBAN1:21
for V being ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for F being ( ( Function-like V30( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) st F : ( ( Function-like V30( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) is Real_homogeneous holds
for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( complex real ext-real ) Real) holds F : ( ( Function-like V30( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . ([**0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) ,r : ( ( ) ( complex real ext-real ) Real) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = r : ( ( ) ( complex real ext-real ) Real) * (F : ( ( Function-like V30( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . (i_FC : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let F be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
attr F is homogeneous means :: HAHNBAN1:def 14
 for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( complex left_complementable right_complementable complementable ) Scalar of ) holds F : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) . (r : ( ( ) ( complex left_complementable right_complementable complementable ) Scalar of ) * v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = |.r : ( ( ) ( complex left_complementable right_complementable complementable ) Scalar of ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) * (F : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) . v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;
end;

definition
let K be ( ( ) ( ) 1-sorted ) ;
let V be ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
let F be ( ( Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ;
attr F is 0-preserving means :: HAHNBAN1:def 15
F : ( ( ) ( right_complementable ) Element of the carrier of K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) . (0. V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) : ( ( ) ( zero ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = 0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) ;
end;

registration
let K be ( ( ) ( ) 1-sorted ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
cluster Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

registration
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster Function-like V30( the carrier of V : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Real_homogeneous -> Function-like V30( the carrier of V : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) 0-preserving for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

definition
let K be ( ( ) ( ) 1-sorted ) ;
let V be ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
func 0RFunctional V -> ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) equals :: HAHNBAN1:def 16
([#] V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) : ( ( ) ( non empty non proper ) Element of bool the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) --> 0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V30( [#] V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty non proper ) Element of bool the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( [#] V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty non proper ) Element of bool the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( [#] V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty non proper ) Element of bool the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Element of bool [:([#] V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) ) : ( ( ) ( non empty non proper ) Element of bool the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty right_complementable add-associative right_zeroed right-distributive ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

registration
let K be ( ( ) ( ) 1-sorted ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
cluster 0RFunctional V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive ;
cluster 0RFunctional V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) 0-preserving ;
end;

registration
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster 0RFunctional V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive 0-preserving ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Real_homogeneous ;
cluster 0RFunctional V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive 0-preserving ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) homogeneous ;
end;

registration
let K be ( ( ) ( ) 1-sorted ) ;
let V be ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ;
cluster V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive 0-preserving for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

registration
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive Real_homogeneous homogeneous for ( ( ) ( ) Element of bool [: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
mode Semi-Norm of V is ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
end;

begin

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
func RealVS V -> ( ( strict ) ( strict ) RLSStruct ) means :: HAHNBAN1:def 17
( addLoopStr(# the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the addF of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( Function-like V30([: the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19([: the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) ) V20( the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) Function-like V30([: the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) , the ZeroF of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( ) Element of the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) addLoopStr ) = addLoopStr(# the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the addF of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( Function-like V30([: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19([: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) ) V20( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) Function-like V30([: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) , the ZeroF of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) addLoopStr ) & ( for r being ( ( ) ( complex real ext-real ) Real)
for v being ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) holds the Mult of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( Function-like V30([:REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ) ( V16() V19([:REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) ) V20( the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) Function-like V30([:REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . (r : ( ( ) ( complex real ext-real ) Real) ,v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) = [**r : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) );
end;

registration
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( strict ) RLSStruct ) -> non empty strict ;
end;

registration
let V be ( ( non empty Abelian ) ( non empty Abelian ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster RealVS V : ( ( non empty Abelian ) ( non empty Abelian ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) -> strict Abelian ;
end;

registration
let V be ( ( non empty add-associative ) ( non empty add-associative ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster RealVS V : ( ( non empty add-associative ) ( non empty add-associative ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) -> strict add-associative ;
end;

registration
let V be ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster RealVS V : ( ( non empty right_zeroed ) ( non empty right_zeroed ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) -> strict right_zeroed ;
end;

registration
let V be ( ( non empty right_complementable ) ( non empty right_complementable ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster RealVS V : ( ( non empty right_complementable ) ( non empty right_complementable ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) -> right_complementable strict ;
end;

registration
let V be ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
cluster RealVS V : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) -> strict vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

theorem :: HAHNBAN1:22
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for M being ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds RealVS M : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) is ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) Subspace of RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) ) ;

theorem :: HAHNBAN1:23
for V being ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for p being ( ( Function-like V30( the carrier of b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds p : ( ( Function-like V30( the carrier of b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) is ( ( Function-like V30( the carrier of (RealVS b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS b1 : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) ;

theorem :: HAHNBAN1:24
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for p being ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) Semi-Norm of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds p : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) Semi-Norm of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) is ( ( Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive positively_homogeneous ) ( V16() V19( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive positively_homogeneous ) Banach-Functional of RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) ) ;

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func projRe l -> ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) means :: HAHNBAN1:def 18
 for i being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds it : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) . i : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = Re (l : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . i : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;
end;

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func projIm l -> ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) means :: HAHNBAN1:def 19
 for i being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds it : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) . i : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = Im (l : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . i : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;
end;

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) ;
func RtoC l -> ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) equals :: HAHNBAN1:def 20
l : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
end;

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func CtoR l -> ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) equals :: HAHNBAN1:def 21
l : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
end;

registration
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive ) ( V16() V19( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive ) Functional of ( ( ) ( non empty ) set ) ) ;
cluster RtoC l : ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive ) ( V16() V19( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive ) Element of bool [: the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive ;
end;

registration
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive ) ( V16() V19( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive ) RFunctional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
cluster CtoR l : ( ( Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive ) ( V16() V19( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive ) Element of bool [: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) -> Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive ;
end;

registration
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) homogeneous ) ( V16() V19( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) homogeneous positively_homogeneous ) Functional of ( ( ) ( non empty ) set ) ) ;
cluster RtoC l : ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) homogeneous ) ( V16() V19( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) homogeneous positively_homogeneous ) Element of bool [: the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) -> Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Real_homogeneous ;
end;

registration
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Real_homogeneous ) ( V16() V19( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Real_homogeneous 0-preserving ) RFunctional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
cluster CtoR l : ( ( Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Real_homogeneous ) ( V16() V19( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Real_homogeneous 0-preserving ) Element of bool [: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) -> Function-like V30( the carrier of (RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) homogeneous ;
end;

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;
func i-shift l -> ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) means :: HAHNBAN1:def 22
 for v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds it : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) . v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) = l : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . (i_FC : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) * v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ;
end;

definition
let V be ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;
let l be ( ( Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty strict ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) ;
func prodReIm l -> ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) Functional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) means :: HAHNBAN1:def 23
 for v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds it : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) . v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = [**((RtoC l : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ,(- ((i-shift (RtoC l : ( ( non empty ) ( non empty ) VectSpStr over V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) RFunctional of V : ( ( non empty ) ( non empty ) VectSpStr over F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ;
end;

theorem :: HAHNBAN1:25
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for l being ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds projRe l : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) is ( ( Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive homogeneous ) ( V16() V19( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive homogeneous positively_homogeneous ) linear-Functional of RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) ) ;

theorem :: HAHNBAN1:26
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for l being ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds projIm l : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ( V16() V19( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) Functional of ( ( ) ( non empty ) set ) ) is ( ( Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive homogeneous ) ( V16() V19( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive homogeneous positively_homogeneous ) linear-Functional of RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) ) ;

theorem :: HAHNBAN1:27
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for l being ( ( Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive homogeneous ) ( V16() V19( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive homogeneous positively_homogeneous ) linear-Functional of RealVS V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) ) holds prodReIm l : ( ( Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) additive homogeneous ) ( V16() V19( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of (RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive additive homogeneous positively_homogeneous ) linear-Functional of RealVS b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() ) RLSStruct ) ) : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) Functional of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) is ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ;

theorem :: HAHNBAN1:28
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) )
for p being ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) Semi-Norm of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for M being ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) )
for l being ( ( Function-like V30( the carrier of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of M : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) st ( for e being ( ( ) ( left_complementable right_complementable complementable ) Vector of ( ( ) ( non empty ) set ) )
for v being ( ( ) ( left_complementable right_complementable complementable ) Vector of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( left_complementable right_complementable complementable ) Vector of ( ( ) ( non empty ) set ) ) = e : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) holds
|.(l : ( ( Function-like V30( the carrier of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) . e : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) <= p : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) Semi-Norm of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . v : ( ( ) ( left_complementable right_complementable complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) holds
ex L being ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) st
( L : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) | the carrier of M : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like ) Element of bool [: the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) = l : ( ( Function-like V30( the carrier of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of b3 : ( ( ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) ) & ( for e being ( ( ) ( left_complementable right_complementable complementable ) Vector of ( ( ) ( non empty ) set ) ) holds |.(L : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) additive homogeneous 0-preserving ) linear-Functional of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . e : ( ( ) ( left_complementable right_complementable complementable ) Vector of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( complex left_complementable right_complementable complementable ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) <= p : ( ( Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) ( V16() V19( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) V20( REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) Function-like V30( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) subadditive homogeneous ) Semi-Norm of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ) . e : ( ( ) ( left_complementable right_complementable complementable ) Vector of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) ) ) ) ;

begin

theorem :: HAHNBAN1:29
for x being ( ( ) ( complex real ext-real ) Real) st x : ( ( ) ( complex real ext-real ) Real) > 0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) holds
for n being ( ( ) ( natural complex real ext-real V33() V34() V192() V193() V194() V195() V196() V197() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) holds (power F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) : ( ( Function-like V30([: the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) :] : ( ( ) ( ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) ( V16() V19([: the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) :] : ( ( ) ( ) set ) ) V20( the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) Function-like V30([: the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) :] : ( ( ) ( ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) Element of bool [:[: the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ,NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) :] : ( ( ) ( ) set ) , the carrier of F_Complex : ( ( strict ) ( non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . ([**x : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ,n : ( ( ) ( natural complex real ext-real V33() V34() V192() V193() V194() V195() V196() V197() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) = [**(x : ( ( ) ( complex real ext-real ) Real) to_power n : ( ( ) ( natural complex real ext-real V33() V34() V192() V193() V194() V195() V196() V197() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( real ) ( complex real ext-real ) set ) ,0 : ( ( ) ( empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() ) Element of NAT : ( ( ) ( V192() V193() V194() V195() V196() V197() V198() ) Element of bool REAL : ( ( ) ( non empty V35() V192() V193() V194() V198() ) set ) : ( ( ) ( ) set ) ) ) **] : ( ( ) ( complex left_complementable right_complementable complementable ) Element of ( ( ) ( non empty non trivial ) set ) ) ;