:: MATRIX12 semantic presentation

begin

definition
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l, k be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
func InterchangeLine (M,l,k) -> ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( ) ( ) 1-sorted ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) , ( ( ) ( ) set ) ) means :: MATRIX12:def 1
( len it : ( ( ) ( ) set ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) = len M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & ( for i, j being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) st i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
( ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) & i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) ) ) );
end;

theorem :: MATRIX12:1
for n, m being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for M1, M2 being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) holds width M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) = width M2 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ;

theorem :: MATRIX12:2
for m, n, l, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for M, M1 being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) )
for i being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) = InterchangeLine (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) holds
( ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) implies Line (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b6 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b6 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) implies Line (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b6 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b6 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) & i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) implies Line (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b6 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b6 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) ) ;

theorem :: MATRIX12:3
for m, n being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) )
for i, j being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for M being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
(a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) * (Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) . j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) : ( ( ) ( ) set ) = a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) * (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( ) M2( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) )) : ( ( ) ( ) M2( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) )) ;

definition
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let a be ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ;
func ScalarXLine (M,l,a) -> ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( ) ( ) 1-sorted ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) , ( ( ) ( ) set ) ) means :: MATRIX12:def 2
( len it : ( ( ) ( ) set ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) = len M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & ( for i, j being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) st i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
( ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = a : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) * (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) ) ) );
end;

theorem :: MATRIX12:4
for n, m, l, i being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) )
for M, M1 being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) = ScalarXLine (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) holds
( ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) implies Line (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) * (Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) implies Line (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) ) ;

definition
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l, k be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let a be ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ;
assume k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ;
func RlineXScalar (M,l,k,a) -> ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( ) ( ) 1-sorted ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) , ( ( ) ( ) set ) ) means :: MATRIX12:def 3
( len it : ( ( ) ( ) M2( bool M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) set ) )) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) = len M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & ( for i, j being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) st i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
( ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) M2( bool M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) set ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = (a : ( ( ) ( ) set ) * (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) + (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) M2( bool M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) set ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) ) ) );
end;

theorem :: MATRIX12:5
for n, m, l, k, i being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) )
for M, M1 being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) = RlineXScalar (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) holds
( ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) implies Line (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b9 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b9 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = (a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) * (Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) + (Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) & ( i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) implies Line (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b9 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b9 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) ) ;

notation
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l, k be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
synonym ILine (M,l,k) for InterchangeLine (M,l,k);
end;

notation
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let a be ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ;
synonym SXLine (M,l,a) for ScalarXLine (M,l,a);
end;

notation
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l, k be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let a be ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ;
synonym RLineXS (M,l,k,a) for RlineXScalar (M,l,k,a);
end;

theorem :: MATRIX12:6
for l, n, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for A being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
(ILine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) * A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = ILine (A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:7
for n being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for l being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) )
for A being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
(SXLine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) * A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = SXLine (A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:8
for l, n, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) )
for A being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
(RLineXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) * A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = RLineXS (A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:9
for n, m, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for M being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) holds ILine (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) = M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:10
for n, m, l, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for M being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) holds ILine (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) = ILine (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:11
for n, m, l, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for M being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
ILine ((ILine (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) = M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:12
for l, n, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
( ILine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) is invertible & (ILine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ~ : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = ILine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) ;

theorem :: MATRIX12:13
for l, n, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) holds
( RLineXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) is invertible & (RLineXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ~ : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = RLineXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,(- a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( ) M2( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) )) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) ;

theorem :: MATRIX12:14
for l, n being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) <> 0. K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V47(b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ) ) M2( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) )) holds
( SXLine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) is invertible & (SXLine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ~ : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = SXLine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,(a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ") : ( ( ) ( ) M2( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) )) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) ;

definition
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l, k be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
assume that
l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) and
k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) and
n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) and
m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ;
func InterchangeCol (M,l,k) -> ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( ) ( ) 1-sorted ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) , ( ( ) ( ) set ) ) means :: MATRIX12:def 4
( len it : ( ( ) ( ) set ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) = len M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & ( for i, j being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) st i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
( ( j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) & ( j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) & ( j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) & j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) ) ) );
end;

definition
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let a be ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ;
assume that
l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) and
n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) and
m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ;
func ScalarXCol (M,l,a) -> ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( ) ( ) 1-sorted ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) , ( ( ) ( ) set ) ) means :: MATRIX12:def 5
( len it : ( ( ) ( ) set ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) = len M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & ( for i, j being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) st i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
( ( j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = a : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) * (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) & ( j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) set ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) ) ) );
end;

definition
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l, k be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let a be ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ;
assume that
l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) and
k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) and
n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) and
m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ;
func RcolXScalar (M,l,k,a) -> ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( ) ( ) 1-sorted ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) , ( ( ) ( ) set ) ) means :: MATRIX12:def 6
( len it : ( ( ) ( ) M2( bool M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) set ) )) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) = len M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & ( for i, j being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) st i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) holds
( ( j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) = l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) M2( bool M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) set ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = (a : ( ( ) ( ) set ) * (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) + (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) & ( j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) implies it : ( ( ) ( ) M2( bool M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) : ( ( ) ( ) set ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) = M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) ) ) );
end;

notation
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l, k be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
synonym ICol (M,l,k) for InterchangeCol (M,l,k);
end;

notation
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let a be ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ;
synonym SXCol (M,l,a) for ScalarXCol (M,l,a);
end;

notation
let n, m be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let K be ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ;
let M be ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;
let l, k be ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ;
let a be ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ;
synonym RColXS (M,l,k,a) for RcolXScalar (M,l,k,a);
end;

theorem :: MATRIX12:15
for l, k, n, m being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for M, M1 being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) = M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) @ : ( ( tabular ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) FinSequence of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) holds
(ILine (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) @ : ( ( tabular ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) FinSequence of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = ICol (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:16
for l, n, m being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) )
for M, M1 being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) = M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) @ : ( ( tabular ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) FinSequence of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) holds
(SXLine (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) @ : ( ( tabular ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) FinSequence of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = SXCol (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:17
for l, k, n, m being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) )
for M, M1 being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) & M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) = M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) @ : ( ( tabular ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) FinSequence of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) holds
(RLineXS (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) @ : ( ( tabular ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) FinSequence of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) = RColXS (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:18
for l, n, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for A being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) holds
A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) * (ICol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = ICol (A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:19
for l, n being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) )
for A being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) holds
A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) * (SXCol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = SXCol (A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:20
for l, n, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) )
for A being ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) holds
A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) * (RColXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = RColXS (A : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:21
for l, n, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) holds
(ICol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ~ : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = ICol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:22
for l, n, k being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) <> l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) & n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) holds
(RColXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ~ : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = RColXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,(- a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( ) M2( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) )) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;

theorem :: MATRIX12:23
for l, n being ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat)
for K being ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) st l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( ) M2( bool NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) & a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) <> 0. K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V47(b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ) ) M2( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) )) & n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) > 0 : ( ( ) ( V11() V21() V22() V23() V25() V26() V27() ext-real non positive non negative ) M3( REAL : ( ( ) ( ) set ) , NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) holds
(SXCol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ~ : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) = SXCol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,(a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ") : ( ( ) ( ) M2( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) )) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ;