:: MATRIX16 semantic presentation

begin

theorem :: MATRIX16:1
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) holds (1_ K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:2
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) holds (- (1_ K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = - p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

definition
let K be ( ( ) ( ) set ) ;
let M be ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) ) ;
let p be ( ( Relation-like Function-like FinSequence-like ) ( Relation-like Function-like FinSequence-like ) FinSequence) ;
pred M is_line_circulant_about p means :: MATRIX16:def 1
( len p : ( ( Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) ( Relation-like [:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) Element of bool [:[:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = width M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & ( for i, j being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) st [i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ] : ( ( ) ( ) set ) in Indices M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ) set ) holds
M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) * (i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ) : ( ( ) ( ) Element of K : ( ( ) ( ) doubleLoopStr ) ) = p : ( ( Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) ( Relation-like [:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) Element of bool [:[:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . (((j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) - i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ) : ( ( ) ( V11() V12() integer ext-real ) set ) mod (len p : ( ( Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) ( Relation-like [:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) Element of bool [:[:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( integer ) ( V11() V12() integer ext-real ) set ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) : ( ( ) ( ) set ) ) );
end;

definition
let K be ( ( ) ( ) set ) ;
let M be ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) ) ;
attr M is line_circulant means :: MATRIX16:def 2
 ex p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( ) ( ) doubleLoopStr ) ) st
( len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( ) ( ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = width M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) is_line_circulant_about p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( ) ( ) set ) ) );
end;

definition
let K be ( ( non empty ) ( non empty ) set ) ;
let p be ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( non empty ) ( non empty ) set ) ) ;
attr p is first-line-of-circulant means :: MATRIX16:def 3
 ex M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) doubleLoopStr ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) doubleLoopStr ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len p : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,K : ( ( ) ( ) doubleLoopStr ) ) st M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,K : ( ( non empty ) ( non empty ) set ) ) is_line_circulant_about p : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) ;
end;

definition
let K be ( ( ) ( ) set ) ;
let M be ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) ) ;
let p be ( ( Relation-like Function-like FinSequence-like ) ( Relation-like Function-like FinSequence-like ) FinSequence) ;
pred M is_col_circulant_about p means :: MATRIX16:def 4
( len p : ( ( Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) ( Relation-like [:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) Element of bool [:[:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & ( for i, j being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) st [i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ] : ( ( ) ( ) set ) in Indices M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ) set ) holds
M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) * (i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ) : ( ( ) ( ) Element of K : ( ( ) ( ) doubleLoopStr ) ) = p : ( ( Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) ( Relation-like [:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) Element of bool [:[:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) . (((i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) - j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ) : ( ( ) ( V11() V12() integer ext-real ) set ) mod (len p : ( ( Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) ( Relation-like [:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) Element of bool [:[:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( integer ) ( V11() V12() integer ext-real ) set ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) : ( ( ) ( ) set ) ) );
end;

definition
let K be ( ( ) ( ) set ) ;
let M be ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) ) ;
attr M is col_circulant means :: MATRIX16:def 5
 ex p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( ) ( ) doubleLoopStr ) ) st
( len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( ) ( ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) is_col_circulant_about p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( ) ( ) set ) ) );
end;

definition
let K be ( ( non empty ) ( non empty ) set ) ;
let p be ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( non empty ) ( non empty ) set ) ) ;
attr p is first-col-of-circulant means :: MATRIX16:def 6
 ex M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) doubleLoopStr ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) doubleLoopStr ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len p : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,K : ( ( ) ( ) doubleLoopStr ) ) st M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,K : ( ( non empty ) ( non empty ) set ) ) is_col_circulant_about p : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) set ) ;
let p be ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( non empty ) ( non empty ) set ) ) ;
assume p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( non empty ) ( non empty ) set ) ) is first-line-of-circulant ;
func LCirc p -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) doubleLoopStr ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) doubleLoopStr ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len p : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,K : ( ( ) ( ) doubleLoopStr ) ) means :: MATRIX16:def 7
it : ( ( Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) ( Relation-like [:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) Element of bool [:[:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) is_line_circulant_about p : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) set ) ;
let p be ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( non empty ) ( non empty ) set ) ) ;
assume p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of K : ( ( non empty ) ( non empty ) set ) ) is first-col-of-circulant ;
func CCirc p -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) doubleLoopStr ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) doubleLoopStr ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len p : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,K : ( ( ) ( ) doubleLoopStr ) ) means :: MATRIX16:def 8
it : ( ( Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) ( Relation-like [:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) -defined K : ( ( ) ( ) doubleLoopStr ) -valued Function-like V27([:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) ) ) Element of bool [:[:K : ( ( ) ( ) doubleLoopStr ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) ,K : ( ( ) ( ) doubleLoopStr ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) is_col_circulant_about p : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) doubleLoopStr ) ) ;
end;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ;
cluster Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like first-line-of-circulant first-col-of-circulant for ( ( ) ( ) FinSequence of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ;
end;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ;
let n be ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ;
cluster 0. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) -> line_circulant col_circulant ;
end;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ;
let n be ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ;
let a be ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ;
cluster (n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) --> a : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( Relation-like Function-like FinSequence-like tabular ) ( Relation-like Function-like FinSequence-like tabular ) set ) -> line_circulant for ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ;
cluster (n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) --> a : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( Relation-like Function-like FinSequence-like tabular ) ( Relation-like Function-like FinSequence-like tabular ) set ) -> col_circulant for ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ;
end;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ;
cluster Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular line_circulant col_circulant for ( ( ) ( ) FinSequence of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) ;
end;

theorem :: MATRIX16:3
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) set )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,D : ( ( non empty ) ( non empty ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) is line_circulant & n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() Function-like functional integer ext-real non positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) @ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) is col_circulant ;

theorem :: MATRIX16:4
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) set )
for t being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of D : ( ( non empty ) ( non empty ) set ) )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,D : ( ( non empty ) ( non empty ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) is_line_circulant_about t : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of b2 : ( ( non empty ) ( non empty ) set ) ) & n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() Function-like functional integer ext-real non positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
t : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of b2 : ( ( non empty ) ( non empty ) set ) ) = Line (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V37( width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of (width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MATRIX16:5
for i, j, n, k, l being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) set )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b6 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b6 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,D : ( ( non empty ) ( non empty ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b6 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b6 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b3 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b6 : ( ( non empty ) ( non empty ) set ) ) is line_circulant & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in [:(Seg n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(Seg n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( ) set ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) & l : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b6 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b6 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b3 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b6 : ( ( non empty ) ( non empty ) set ) ) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b6 : ( ( non empty ) ( non empty ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b6 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b6 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b3 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b6 : ( ( non empty ) ( non empty ) set ) ) * (k : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,l : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b6 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MATRIX16:6
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:7
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) + M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:8
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
(M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) + M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:9
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:10
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:11
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
- M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:12
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) - M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:13
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:14
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:15
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:16
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is line_circulant holds
((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:17
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) set )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,D : ( ( non empty ) ( non empty ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) is col_circulant & n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() Function-like functional integer ext-real non positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) @ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) is line_circulant ;

theorem :: MATRIX16:18
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) set )
for t being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of D : ( ( non empty ) ( non empty ) set ) )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,D : ( ( non empty ) ( non empty ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) is_col_circulant_about t : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of b2 : ( ( non empty ) ( non empty ) set ) ) & n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() Function-like functional integer ext-real non positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
t : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of b2 : ( ( non empty ) ( non empty ) set ) ) = Col (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V37( len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of (len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b2 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MATRIX16:19
for i, j, n, k, l being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) set )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b6 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b6 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,D : ( ( non empty ) ( non empty ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b6 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b6 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b3 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b6 : ( ( non empty ) ( non empty ) set ) ) is col_circulant & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in [:(Seg n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(Seg n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( ) set ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) & l : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b6 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b6 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b3 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b6 : ( ( non empty ) ( non empty ) set ) ) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b6 : ( ( non empty ) ( non empty ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b6 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b6 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b3 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b6 : ( ( non empty ) ( non empty ) set ) ) * (k : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,l : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b6 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MATRIX16:20
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:21
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) + M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:22
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
(M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) + M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:23
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:24
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:25
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
- M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:26
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) - M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:27
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:28
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:29
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:30
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b, c being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1, M2, M3 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant & M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is col_circulant holds
((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) - (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (c : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:31
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant holds
- p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant ;

theorem :: MATRIX16:32
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant holds
LCirc (- p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (- b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = - (LCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:33
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant ;

theorem :: MATRIX16:34
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant holds
LCirc (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = (LCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (LCirc q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX16:35
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant holds
- p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant ;

theorem :: MATRIX16:36
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant holds
CCirc (- p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (- b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = - (CCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:37
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant ;

theorem :: MATRIX16:38
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant holds
CCirc (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = (CCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (CCirc q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX16:39
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() Function-like functional integer ext-real non positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is col_circulant ;

theorem :: MATRIX16:40
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() Function-like functional integer ext-real non positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is line_circulant ;

theorem :: MATRIX16:41
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant holds
a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant ;

theorem :: MATRIX16:42
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant holds
LCirc (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (LCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:43
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (LCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (LCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = LCirc ((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) + b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len ((b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) + b3 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:44
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (LCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (LCirc q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) = LCirc (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:45
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-circulant & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (LCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (LCirc q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) = LCirc ((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len ((b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b3 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:46
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant holds
a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant ;

theorem :: MATRIX16:47
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant holds
CCirc (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (CCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:48
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (CCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (CCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = CCirc ((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) + b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len ((b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) + b3 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:49
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() Function-like functional integer ext-real non positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (CCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (CCirc q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) = CCirc (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:50
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-col-of-circulant & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (CCirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (CCirc q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) = CCirc ((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len ((b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b3 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

notation
let K be ( ( ) ( ) set ) ;
let M be ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined K : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of ( ( ) ( functional FinSequence-membered ) FinSequenceSet of K : ( ( ) ( ) set ) ) ) ;
synonym circulant M for line_circulant ;
end;

begin

definition
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ;
let M1 be ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of ) ;
let p be ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ;
pred M1 is_anti-circular_about p means :: MATRIX16:def 9
( len p : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = width M1 : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & ( for i, j being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) st [i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ] : ( ( ) ( ) set ) in Indices M1 : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) & i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) <= j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) holds
M1 : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) * (i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ) : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = p : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) . (((j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) - i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ) : ( ( ) ( V11() V12() integer ext-real ) set ) mod (len p : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( integer ) ( V11() V12() integer ext-real ) set ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) : ( ( ) ( ) set ) ) & ( for i, j being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) st [i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ] : ( ( ) ( ) set ) in Indices M1 : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) & i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) >= j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) holds
M1 : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) * (i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ) : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) = (- p : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) . (((j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) - i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ) : ( ( ) ( V11() V12() integer ext-real ) set ) mod (len p : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( integer ) ( V11() V12() integer ext-real ) set ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) : ( ( ) ( ) set ) ) );
end;

definition
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ;
let M be ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of ) ;
attr M is anti-circular means :: MATRIX16:def 10
 ex p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st
( len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = width M : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) is_anti-circular_about p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) );
end;

definition
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ;
let p be ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ;
attr p is first-line-of-anti-circular means :: MATRIX16:def 11
 ex M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len p : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is_anti-circular_about p : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;
end;

definition
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ;
let p be ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ;
assume p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular ;
func ACirc p -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len p : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) means :: MATRIX16:def 12
it : ( ( ) ( ) Element of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) is_anti-circular_about p : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;
end;

theorem :: MATRIX16:51
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for M1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is anti-circular holds
a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is anti-circular ;

theorem :: MATRIX16:52
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is anti-circular & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is anti-circular holds
M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) + M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is anti-circular ;

theorem :: MATRIX16:53
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field)
for n, i, j being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat)
for M1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) , ( ( ) ( non empty non trivial ) set ) ) st [i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ] : ( ( ) ( ) set ) in Indices M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) & i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) = j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) & M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) , ( ( ) ( non empty non trivial ) set ) ) is anti-circular holds
M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ,j : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( V49(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() Fanoian ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital Fanoian ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:54
for i, j, n, k, l being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b3 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is anti-circular & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in [:(Seg n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(Seg n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( ) set ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) & l : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() integer ext-real ) set ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b3 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (k : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,l : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b3 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b6 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:55
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is anti-circular holds
- M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is anti-circular ;

theorem :: MATRIX16:56
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for M1, M2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is anti-circular & M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is anti-circular holds
M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) - M2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is anti-circular ;

theorem :: MATRIX16:57
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) )
for M1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is_anti-circular_about p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) & n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() Function-like functional integer ext-real non positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) = Line (M1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() integer ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V37( width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of (width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX16:58
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular holds
- p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular ;

theorem :: MATRIX16:59
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular holds
ACirc (- p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (- b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = - (ACirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:60
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular ;

theorem :: MATRIX16:61
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
ACirc (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = (ACirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) + (ACirc q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX16:62
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular holds
a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular ;

theorem :: MATRIX16:63
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular holds
ACirc (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (ACirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:64
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (ACirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (ACirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = ACirc ((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) + b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len ((b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) + b3 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:65
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (ACirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (ACirc q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) = ACirc (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len (b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX16:66
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) is first-line-of-anti-circular & len p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (ACirc p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (ACirc q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , len b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) = ACirc ((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular ) Matrix of len ((b2 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + (b3 : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * b5 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like FinSequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ;

registration
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) Field) ;
let n be ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ;
cluster 0. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like FinSequence-like Function-yielding V105() tabular line_circulant col_circulant ) Matrix of n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() integer ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V114() V116() V119() V120() V121() well-unital V133() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible unital V114() V116() V119() V120() V121() right-distributive left-distributive right_unital well-unital V133() left_unital ) doubleLoopStr ) : ( ( ) ( non empty non trivial ) set ) ) -> anti-circular ;
end;