:: MATRTOP2 semantic presentation

begin

theorem :: MATRTOP2:1
for X being ( ( ) ( ) set )
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) holds
( X : ( ( ) ( ) set ) is ( ( ) ( Relation-like the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) iff X : ( ( ) ( ) set ) is ( ( ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of TOP-REAL n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) ) ;

theorem :: MATRTOP2:2
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for Lv being ( ( ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) )
for Lr being ( ( ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of TOP-REAL n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) st Lr : ( ( ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) = Lv : ( ( ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) holds
Carrier Lr : ( ( ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( finite ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = Carrier Lv : ( ( ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( finite ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: MATRTOP2:3
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for F being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for fr being ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) )
for Fv being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for fv being ( ( Function-like quasi_total ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) st fr : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) = fv : ( ( Function-like quasi_total ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) & F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) = Fv : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) holds
fr : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) (#) F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) = fv : ( ( Function-like quasi_total ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) (#) Fv : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MATRTOP2:4
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for F being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) )
for Fv being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st Fv : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) = F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) holds
Sum F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) = Sum Fv : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MATRTOP2:5
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for Lv being ( ( ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) )
for Lr being ( ( ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of TOP-REAL n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) st Lr : ( ( ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) = Lv : ( ( ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) holds
Sum Lr : ( ( ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) = Sum Lv : ( ( ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MATRTOP2:6
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for Af being ( ( ) ( ) Subset of )
for Ar being ( ( ) ( ) Subset of ) st Af : ( ( ) ( ) Subset of ) = Ar : ( ( ) ( ) Subset of ) holds
[#] (Lin Ar : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty non proper ) Element of K19( the carrier of (Lin b3 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = [#] (Lin Af : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty non proper ) Element of K19( the carrier of (Lin b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: MATRTOP2:7
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for Af being ( ( ) ( ) Subset of )
for Ar being ( ( ) ( ) Subset of ) st Af : ( ( ) ( ) Subset of ) = Ar : ( ( ) ( ) Subset of ) holds
( Af : ( ( ) ( ) Subset of ) is linearly-independent iff Ar : ( ( ) ( ) Subset of ) is linearly-independent ) ;

theorem :: MATRTOP2:8
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) )
for L being ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) holds L : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) | the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like ) Element of K19(K20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is ( ( ) ( Relation-like the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ) ;

theorem :: MATRTOP2:9
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field)
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) )
for A being ( ( linearly-independent ) ( linearly-independent ) Subset of )
for L1, L2 being ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) st Carrier L1 : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) : ( ( finite ) ( finite ) Element of K19( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= A : ( ( linearly-independent ) ( linearly-independent ) Subset of ) & Carrier L2 : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) : ( ( finite ) ( finite ) Element of K19( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= A : ( ( linearly-independent ) ( linearly-independent ) Subset of ) & Sum L1 : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) = Sum L2 : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) holds
L1 : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) = L2 : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ;

theorem :: MATRTOP2:10
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) )
for L being ( ( ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ) holds L : ( ( ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ) | the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ) : ( ( ) ( non empty ) set ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K19(K20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( Relation-like non empty V12() non finite complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) is ( ( ) ( Relation-like the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ) ) ;

theorem :: MATRTOP2:11
for X being ( ( ) ( ) set )
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for U being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) st [#] U : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty non proper ) Element of K19( the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = [#] W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty non proper ) Element of K19( the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( X : ( ( ) ( ) set ) is ( ( ) ( Relation-like the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of U : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) ) iff X : ( ( ) ( ) set ) is ( ( ) ( Relation-like the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) ) ) ;

theorem :: MATRTOP2:12
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for U being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) )
for LU being ( ( ) ( Relation-like the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of U : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) )
for LW being ( ( ) ( Relation-like the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) ) st LU : ( ( ) ( Relation-like the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) ) = LW : ( ( ) ( Relation-like the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) ) holds
( Carrier LU : ( ( ) ( Relation-like the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) ) : ( ( finite ) ( finite ) Element of K19( the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = Carrier LW : ( ( ) ( Relation-like the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) ) : ( ( ) ( finite ) Element of K19( the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Sum LU : ( ( ) ( Relation-like the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) = Sum LW : ( ( ) ( Relation-like the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) ) ) ;

registration
let m be ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ;
let K be ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ;
let A be ( ( ) ( ) Subset of ) ;
cluster Lin A : ( ( ) ( ) Element of K19( the carrier of (m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) set ) -VectSp_over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) set ) -VectSp_over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) -> strict finite-dimensional ;
end;

begin

theorem :: MATRTOP2:13
for m, n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) st the_rank_of M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) = n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) holds
M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) is ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (Lin (lines b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of Lin (lines M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) ) ;

theorem :: MATRTOP2:14
for K being ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field)
for V, W being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of K : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) )
for T being ( ( Function-like quasi_total V157(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) V200(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V157(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) V200(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ) linear-transformation of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,W : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) )
for A being ( ( ) ( ) Subset of )
for L being ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of A : ( ( ) ( ) Subset of ) ) st T : ( ( Function-like quasi_total V157(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) V200(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V157(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) V200(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ) linear-transformation of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) | A : ( ( ) ( ) Subset of ) : ( ( Function-like ) ( Relation-like b5 : ( ( ) ( ) Subset of ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K19(K20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is one-to-one holds
T : ( ( Function-like quasi_total V157(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) V200(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V157(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) V200(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ) linear-transformation of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) . (Sum L : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b5 : ( ( ) ( ) Subset of ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) = Sum (T : ( ( Function-like quasi_total V157(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) V200(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total V157(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) V200(b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) ) linear-transformation of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ,b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) @ L : ( ( ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b5 : ( ( ) ( ) Subset of ) ) ) : ( ( ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) : ( ( ) ( non empty V12() ) set ) -valued Function-like total quasi_total ) Linear_Combination of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) VectSp of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed V141() V143() well-unital V149() ) ( non empty non degenerated V62() right_complementable almost_left_invertible Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) Field) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MATRTOP2:15
for m, n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for f being ( ( Relation-like Function-like b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) )
for S being ( ( ) ( finite V183() V184() V185() V186() V187() V188() ) Subset of ( ( ) ( finite V38() ) set ) ) st M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) | S : ( ( ) ( finite V183() V184() V185() V186() V187() V188() ) Subset of ( ( ) ( finite V38() ) set ) ) : ( ( Function-like ) ( Relation-like b5 : ( ( ) ( finite V183() V184() V185() V186() V187() V188() ) Subset of ( ( ) ( finite V38() ) set ) ) -defined NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSubsequence-like ) Element of K19(K20(NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ,( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) *) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( Relation-like non empty V12() non finite ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) is one-to-one & rng (M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) | S : ( ( ) ( finite V183() V184() V185() V186() V187() V188() ) Subset of ( ( ) ( finite V38() ) set ) ) ) : ( ( Function-like ) ( Relation-like b5 : ( ( ) ( finite V183() V184() V185() V186() V187() V188() ) Subset of ( ( ) ( finite V38() ) set ) ) -defined NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSubsequence-like ) Element of K19(K20(NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ,( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) *) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( Relation-like non empty V12() non finite ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) : ( ( ) ( functional finite FinSequence-membered ) Element of K19(( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) *) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( ) set ) ) = lines M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex L being ( ( ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of lines M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) st
( Sum L : ( ( ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of lines b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = (Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . f : ( ( Relation-like Function-like b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) set ) & ( for k being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) st k : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) in S : ( ( ) ( finite V183() V184() V185() V186() V187() V188() ) Subset of ( ( ) ( finite V38() ) set ) ) holds
L : ( ( ) ( Relation-like the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of lines b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) . (Line (M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ,k : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like finite width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -element FinSequence-like FinSubsequence-like ) Element of (width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -tuples_on the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( ) set ) = Sum (Seq (f : ( ( Relation-like Function-like b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) | (M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) " {(Line (M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ,k : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like finite width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -element FinSequence-like FinSubsequence-like ) Element of (width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -tuples_on the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) } : ( ( ) ( functional non empty V12() finite V38() 1 : ( ( ) ( non empty V23() V27() V28() V29() V30() V31() finite cardinal ext-real positive non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -element ) set ) ) : ( ( ) ( finite V183() V184() V185() V186() V187() V188() ) Element of K19(NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) : ( ( Relation-like ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) " {(Line (b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ,b7 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like finite width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -element FinSequence-like FinSubsequence-like ) Element of (width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -tuples_on the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) } : ( ( ) ( functional non empty V12() finite V38() 1 : ( ( ) ( non empty V23() V27() V28() V29() V30() V31() finite cardinal ext-real positive non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -element ) set ) : ( ( ) ( finite V183() V184() V185() V186() V187() V188() ) Element of K19(NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) set ) : ( ( V28() ) ( V28() V29() ext-real ) set ) ) ) ;

theorem :: MATRTOP2:16
for n, m being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for f being ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) st M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) is without_repeated_line holds
ex L being ( ( ) ( Relation-like the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of lines M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) st
( Sum L : ( ( ) ( Relation-like the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of lines b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) = (Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . f : ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) set ) & ( for k being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) st k : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) in dom f : ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) : ( ( ) ( finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element V183() V184() V185() V186() V187() V188() ) Element of K19(NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) : ( ( ) ( non empty V12() non finite ) set ) ) holds
L : ( ( ) ( Relation-like the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like total quasi_total ) Linear_Combination of lines b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) . (Line (M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ,k : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like finite width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -element FinSequence-like FinSubsequence-like ) Element of (width b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -tuples_on the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( ) set ) = f : ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) . k : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( ) ( V28() V29() ext-real ) set ) ) ) ;

theorem :: MATRTOP2:17
for n, m being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for f being ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) )
for B being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (Lin (lines b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of Lin (lines M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) ) st B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (Lin (lines b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of Lin (lines b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) ) = M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) holds
for Mf being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st Mf : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = (Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . f : ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) set ) holds
Mf : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) |-- B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (Lin (lines b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of Lin (lines b4 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) = f : ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) ;

theorem :: MATRTOP2:18
for n, m being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) holds rng (Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of K19( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = [#] (Lin (lines M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty non proper ) Element of K19( the carrier of (Lin (lines b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: MATRTOP2:19
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for F being ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st rng F : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( finite ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is linearly-independent holds
ex M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) st
( M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) is invertible & M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) | (len F : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) = F : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) ;

theorem :: MATRTOP2:20
for n, k being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for f being ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence)
for B being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) st B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) = MX2FinS (1. (F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ,n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) holds
( f : ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) in Lin (rng (B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) | k : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) iff f : ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) = (f : ( ( Relation-like Function-like b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) | k : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) set ) ^ ((n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -' k : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) |-> 0 : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) ) : ( ( ) ( Relation-like empty-yielding NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -valued Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -' b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued ) Element of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -' b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) -tuples_on NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) ) : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) set ) ) ;

theorem :: MATRTOP2:21
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for F being ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) st rng F : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( finite ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is linearly-independent holds
for B being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) st B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) = MX2FinS (1. (F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ,n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) holds
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) st M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) is invertible & M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) | (len F : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) = F : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) holds
(Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) .: ([#] (Lin (rng (B : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) | (len F : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) ) : ( ( ) ( non empty non proper ) Element of K19( the carrier of (Lin (rng (b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) OrdBasis of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) | (len b2 : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) Subspace of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -VectSp_over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional ) VectSpStr over F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = [#] (Lin (rng F : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty non proper ) Element of K19( the carrier of (Lin (rng b2 : ( ( one-to-one ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( finite ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: MATRTOP2:22
for n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for A, B being ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) st card A : ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) : ( ( cardinal ) ( V23() cardinal ) set ) = card B : ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) : ( ( cardinal ) ( V23() cardinal ) set ) holds
ex M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) st
( M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) is invertible & (Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) .: ([#] (Lin A : ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) ) : ( ( ) ( non empty non proper ) Element of K19( the carrier of (Lin b2 : ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = [#] (Lin B : ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty non proper ) Element of K19( the carrier of (Lin b3 : ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) ) : ( ( strict ) ( non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

begin

theorem :: MATRTOP2:23
for m, n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) )
for A being ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) st the_rank_of M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) = n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) holds
(Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) .: A : ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is linearly-independent ;

theorem :: MATRTOP2:24
for m, n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) )
for A being ( ( affinely-independent ) ( affinely-independent ) Subset of ) st the_rank_of M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) = n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) holds
(Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) .: A : ( ( affinely-independent ) ( affinely-independent ) Subset of ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is affinely-independent ;

theorem :: MATRTOP2:25
for m, n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) )
for A being ( ( affinely-independent ) ( affinely-independent ) Subset of ) st the_rank_of M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) = n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) holds
for v being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of ( ( ) ( non empty ) set ) ) st v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of ( ( ) ( non empty ) set ) ) in Affin A : ( ( affinely-independent ) ( affinely-independent ) Subset of ) : ( ( ) ( V268( TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) ) Element of K19( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( (Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) in Affin ((Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) .: A : ( ( affinely-independent ) ( affinely-independent ) Subset of ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V268( TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) ) ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for f being ( ( Relation-like Function-like b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) holds (v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of ( ( ) ( non empty ) set ) ) |-- A : ( ( affinely-independent ) ( affinely-independent ) Subset of ) ) : ( ( ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of b4 : ( ( affinely-independent ) ( affinely-independent ) Subset of ) ) . f : ( ( Relation-like Function-like b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) : ( ( ) ( V28() V29() ext-real ) set ) = (((Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) |-- ((Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) .: A : ( ( affinely-independent ) ( affinely-independent ) Subset of ) ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Linear_Combination of (Mx2Tran b3 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) .: b4 : ( ( affinely-independent ) ( affinely-independent ) Subset of ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) . ((Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . f : ( ( Relation-like Function-like b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined Function-like finite b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( V28() V29() ext-real ) set ) ) ) ;

theorem :: MATRTOP2:26
for m, n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) )
for A being ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) st the_rank_of M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) = n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) holds
(Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) " A : ( ( linearly-independent ) ( affinely-independent linearly-independent ) Subset of ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is linearly-independent ;

theorem :: MATRTOP2:27
for m, n being ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat)
for M being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,m : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) )
for A being ( ( affinely-independent ) ( affinely-independent ) Subset of ) st the_rank_of M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) : ( ( ) ( V23() V27() V28() V29() V30() V31() finite cardinal ext-real non negative V183() V184() V185() V186() V187() V188() ) Element of NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) ) = n : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) holds
(Mx2Tran M : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V12() V23() non finite cardinal limit_cardinal V183() V184() V185() V186() V187() V188() V189() ) Element of K19(REAL : ( ( ) ( non empty V12() non finite V183() V184() V185() V189() ) set ) ) : ( ( ) ( non empty V12() non finite ) set ) ) -defined the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) * : ( ( ) ( functional non empty FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V62() right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V139() V141() V143() right-distributive left-distributive right_unital well-unital V149() left_unital ) doubleLoopStr ) : ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) , ( ( ) ( non empty V12() V183() V184() V185() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -defined the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Element of K19(K20( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL b1 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) " A : ( ( affinely-independent ) ( affinely-independent ) Subset of ) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL b2 : ( ( V27() ) ( V23() V27() V28() V29() finite cardinal ext-real non negative ) Nat) ) : ( ( V231() ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V231() V237() V238() ) L18()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is affinely-independent ;