:: MATRIX14 semantic presentation

begin

definition
let K be ( ( non empty ) ( non empty ) ZeroStr ) ;
let n be ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ;
func 0* (K,n) -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( ) set ) ) equals :: MATRIX14:def 1
n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) |-> (0. K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like V28() V35(n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) FinSequence-like FinSubsequence-like ) Element of n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) -tuples_on the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ;
end;

definition
let K be ( ( non empty ) ( non empty ) ZeroStr ) ;
let n be ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ;
:: original: 0*
redefine func 0* (K,n) -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like V28() V35(n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) FinSequence-like FinSubsequence-like ) Element of n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) -tuples_on the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ;
end;

theorem :: MATRIX14:1
for L being ( ( non empty ) ( non empty ) addLoopStr )
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) holds x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) is ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() V35( len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of L : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MATRIX14:2
for L being ( ( non empty ) ( non empty ) addLoopStr )
for x1, x2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
len (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) + x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: MATRIX14:3
for L being ( ( non empty ) ( non empty ) addLoopStr )
for x1, x2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
len (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: MATRIX14:4
for G being ( ( non empty ) ( non empty ) multLoopStr )
for x1, x2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) in dom (mlt (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V28() ) set ) ) holds
( (mlt (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) . i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) * (x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) & (mlt (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) /. i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) = (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) * (x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MATRIX14:5
for L being ( ( non empty ) ( non empty ) addLoopStr )
for x1, x2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for i being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) + x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) . i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) : ( ( ) ( ) set ) = (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) + (x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) & (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) . i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) : ( ( ) ( ) set ) = (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) - (x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) /. i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MATRIX14:6
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) holds
( - (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = (- a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & - (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (- x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX14:7
for G being ( ( non empty ) ( non empty ) multLoopStr )
for x1, x2, y1, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
mlt ((x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ^ y1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) ,(x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ^ y2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) = (mlt (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) ^ (mlt (y1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multLoopStr ) : ( ( ) ( non empty ) set ) ) ;

notation
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ;
let e1, e2 be ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ;
synonym |(e1,e2)| for e1 "*" e2;
end;

theorem :: MATRIX14:8
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) )
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( mlt ((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (mlt (x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & mlt (x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (mlt (x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX14:9
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) )
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
|((a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * |(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:10
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) )
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
|(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,(a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * |(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:11
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x, y1, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) )
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
|(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,(y1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) + y2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = |(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + |(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:12
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x1, x2, y1, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
|((x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ^ y1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,(x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ^ y2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = |(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,x2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) + |(y1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:13
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for p1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() V35(b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) holds mlt (p1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() V35(b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ,(n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) |-> (0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() V35(b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() V35(b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) = n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) |-> (0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() V35(b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

notation
let n be ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ;
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ;
let A be ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ;
synonym Inv A for A ~ ;
end;

begin

theorem :: MATRIX14:14
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) holds
( 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = {} : ( ( ) ( ) set ) ) ;

theorem :: MATRIX14:15
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds
( A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = {} : ( ( ) ( ) set ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = 0. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX14:16
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible ;

theorem :: MATRIX14:17
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A, B, C being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * C : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * C : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:18
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A, B being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds
( A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ~ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) iff ( B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: MATRIX14:19
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds
( A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible iff ex B being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st
( B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: MATRIX14:20
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) holds |(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,(0* (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,(len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() V35( len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:21
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) holds |((0* (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,(len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() V35( len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (len b2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:22
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) holds |(<*(0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like non empty trivial V28() V35(1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,<*a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like non empty trivial V28() V35(1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

definition
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ;
let n, i be ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ;
func Base_FinSeq (K,n,i) -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( ) set ) ) equals :: MATRIX14:def 2
 Replace ((n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) |-> (0. K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like V28() V35(n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) FinSequence-like FinSubsequence-like ) Element of n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) -tuples_on the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ,i : ( ( Function-like V18([:n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) ,(1. K : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( ) Element of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

theorem :: MATRIX14:23
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for n, i being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) holds len (Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = n : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ;

theorem :: MATRIX14:24
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for i, n being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) holds
(Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) . i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) : ( ( ) ( ) set ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:25
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for i, j, n being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <> j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) holds
(Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) . j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) : ( ( ) ( ) set ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:26
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for i, n being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) holds
(1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) . i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined Function-like V28() FinSequence-like FinSubsequence-like ) set ) = Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:27
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for i, j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = (Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) . j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: MATRIX14:28
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds
( A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = 0. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) iff for i, j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX14:29
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds
( A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) iff for i, j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = IFEQ (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,(1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ,(0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

begin

theorem :: MATRIX14:30
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A, B being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) @ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = (B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) @) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) @) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:31
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible holds
( A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) @ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) is invertible & (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) @) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ~ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ~) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) @ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX14:32
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) )
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) st ex i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) . i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) & ( for j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <> i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) . j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) holds
Sum x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:33
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for f1, f2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) holds
( dom (mlt (f1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,f2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V28() ) set ) ) = (dom f1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V28() ) set ) ) /\ (dom f2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V28() ) set ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V28() ) set ) ) & ( for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) in dom (mlt (f1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,f2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V28() ) set ) ) holds
(mlt (f1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,f2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) . i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = (f1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) /. i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (f2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) /. i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: MATRIX14:34
for m being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) )
for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) = mlt (x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,(Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) . i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) . i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & ( for j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <> i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) . j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: MATRIX14:35
for m, i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( |(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,(Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) . i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & |(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,(Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) /. i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX14:36
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for m, i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
|((Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ,(Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) )| : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:37
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for P, Q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) <> 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) " : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & ( for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) . i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined Function-like V28() FinSequence-like FinSubsequence-like ) set ) = Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) & Q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) & ( for j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
Q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = - (a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) * (P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
Q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) . i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined Function-like V28() FinSequence-like FinSubsequence-like ) set ) = Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) holds
( P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & Q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ~ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX14:38
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) )
for P being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) <> 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty non trivial ) set ) ) " : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & ( for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) . i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined Function-like V28() FinSequence-like FinSubsequence-like ) set ) = Base_FinSeq (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) -valued Function-like V28() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty non trivial ) set ) ) ) holds
P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible ;

theorem :: MATRIX14:39
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) <> 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) holds
ex P being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st
( P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & ( for j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) holds
(A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: MATRIX14:40
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) <> 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) holds
ex P being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st
( P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & (P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & ( for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( for j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) holds
(P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: MATRIX14:41
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) <> 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) holds
ex P, Q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st
( P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & Q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & ((P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * Q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & ( for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
((P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * Q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( for j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
((P : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * Q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

begin

theorem :: MATRIX14:42
for D being ( ( non empty ) ( non empty ) set )
for m, n, i, j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,D : ( ( non empty ) ( non empty ) set ) ) holds Swap (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b3 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) is ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of m : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,D : ( ( non empty ) ( non empty ) set ) ) ;

definition
let K be ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ;
let n be ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ;
let i0 be ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ;
func SwapDiagonal (K,n,i0) -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) , ( ( ) ( ) set ) ) equals :: MATRIX14:def 3
 Swap ((1. (K : ( ( ) ( ) 1-sorted ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) , the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( Function-like V18([:n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) ( Relation-like Function-like V18([:n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ) ) Element of bool [:[:n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) ,n : ( ( ) ( ) VectSpStr over K : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() ) FinSequence of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ;
end;

theorem :: MATRIX14:43
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for i0 being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds
for i, j being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <> 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( ( ( not i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & not i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) or ( not j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & not j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) ) implies ( ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <> j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ) ) ;

theorem :: MATRIX14:44
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) )
for i being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:45
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) )
for i, j being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <> j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) holds
(SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:46
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for n, i0 being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & ( for i, j being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <> j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ) holds
A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:47
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for n, i0 being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <> 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & ( for i, j being ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( ( ( not i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & not i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) or ( not j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & not j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) ) implies ( ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) = j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) <> j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) implies A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ) ) ) holds
A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b2 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:48
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) )
for i0 being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( ( for j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( ((SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & ((SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) & ( for i, j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <> 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <> i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
((SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: MATRIX14:49
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for i0 being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & (SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ~ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX14:50
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for i0 being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) @ : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,i0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:51
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) )
for j0 being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( ( for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & (A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) & ( for i, j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <> 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <> j0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
(A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (SwapDiagonal (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j0 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: MATRIX14:52
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds
( A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) = 0. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) iff for i, j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

begin

theorem :: MATRIX14:53
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) <> 0. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) holds
ex B, C being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st
( B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & C : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & ((B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * C : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) & ( for i being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
((B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * C : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (i : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) & ( for j being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
((B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * C : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) * (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 0. K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( V83(b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ) ) Element of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: MATRIX14:54
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A, B being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) holds
ex B2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * B2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: MATRIX14:55
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) holds
( ex B1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st B1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) iff ex B2 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * B2 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: MATRIX14:56
for n being ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field)
for A, B being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) st A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) * B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) = 1. (K : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) ,n : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) holds
( A : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible & B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V154() V155() V156() V186() V188() well-unital V196() ) ( non empty non degenerated non trivial right_complementable almost_left_invertible V154() V155() V156() unital V186() V188() right-distributive left-distributive right_unital well-unital V196() left_unital ) Field) : ( ( ) ( non empty non trivial ) set ) ) -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular ) Matrix of b1 : ( ( ) ( V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V28() V33() V34() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty non trivial ) set ) ) is invertible ) ;