:: MATRIX10 semantic presentation

begin

definition
let M be ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
attr M is Positive means :: MATRIX10:def 1
 for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) st [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M : ( ( ) ( ) L1()) : ( ( ) ( ) set ) holds
M : ( ( ) ( ) L1()) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) > 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ;
attr M is Negative means :: MATRIX10:def 2
 for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) st [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M : ( ( ) ( ) L1()) : ( ( ) ( ) set ) holds
M : ( ( ) ( ) L1()) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) < 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ;
attr M is Nonpositive means :: MATRIX10:def 3
 for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) st [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M : ( ( ) ( ) L1()) : ( ( ) ( ) set ) holds
M : ( ( ) ( ) L1()) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ;
attr M is Nonnegative means :: MATRIX10:def 4
 for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) st [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M : ( ( ) ( ) L1()) : ( ( ) ( ) set ) holds
M : ( ( ) ( ) L1()) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ;
end;

definition
let M1, M2 be ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
pred M1 is_less_than M2 means :: MATRIX10:def 5
 for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) st [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M1 : ( ( ) ( ) L1()) : ( ( ) ( ) set ) holds
M1 : ( ( ) ( ) L1()) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) < M2 : ( ( ) ( ) VectSpStr over M1 : ( ( ) ( ) L1()) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
pred M1 is_less_or_equal_with M2 means :: MATRIX10:def 6
 for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) st [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M1 : ( ( ) ( ) L1()) : ( ( ) ( ) set ) holds
M1 : ( ( ) ( ) L1()) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= M2 : ( ( ) ( ) VectSpStr over M1 : ( ( ) ( ) L1()) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
end;

definition
let M be ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
func |:M:| -> ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) means :: MATRIX10:def 7
( len it : ( ( ) ( ) VectSpStr over M : ( ( ) ( ) L1()) ) : ( ( ) ( ext-real non negative V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) = len M : ( ( ) ( ) L1()) : ( ( ) ( ext-real non negative V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & width it : ( ( ) ( ) VectSpStr over M : ( ( ) ( ) L1()) ) : ( ( ) ( ext-real non negative V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) = width M : ( ( ) ( ) L1()) : ( ( ) ( ext-real non negative V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & ( for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) st [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M : ( ( ) ( ) L1()) : ( ( ) ( ) set ) holds
it : ( ( ) ( ) VectSpStr over M : ( ( ) ( ) L1()) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) = abs (M : ( ( ) ( ) L1()) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) )) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ) );
end;

definition
let n be ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ;
let M be ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
:: original: -
redefine func - M -> ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( ) ( ) L1()) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
end;

definition
let n be ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ;
let M1, M2 be ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
:: original: +
redefine func M1 + M2 -> ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( ) ( ) L1()) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
end;

definition
let n be ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ;
let M1, M2 be ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
:: original: -
redefine func M1 - M2 -> ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( ) ( ) L1()) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
end;

definition
let n be ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ;
let a be ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
let M be ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
:: original: *
redefine func a * M -> ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( ) ( ) L1()) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
end;

registration
cluster (1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ) --> 1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) : ( ( V16() V21() FinSequence-like tabular ) ( V16() V21() FinSequence-like tabular ) set ) -> tabular Positive for ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
cluster (1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ) --> 1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) : ( ( V16() V21() FinSequence-like tabular ) ( V16() V21() FinSequence-like tabular ) set ) -> tabular Nonnegative for ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
cluster (1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ) --> (- 1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non positive V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( V16() V21() FinSequence-like tabular ) ( V16() V21() FinSequence-like tabular ) set ) -> tabular Negative for ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
cluster (1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ) --> (- 1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non positive V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( V16() V21() FinSequence-like tabular ) ( V16() V21() FinSequence-like tabular ) set ) -> tabular Nonpositive for ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
end;

registration
cluster V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Positive Nonnegative for ( ( ) ( ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
cluster V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Negative Nonpositive for ( ( ) ( ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
end;

registration
let M be ( ( tabular Positive ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Positive ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
cluster M : ( ( tabular Positive ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Positive ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) @ : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) -> tabular Positive ;
end;

registration
let M be ( ( tabular Negative ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Negative ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
cluster M : ( ( tabular Negative ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Negative ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) @ : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) -> tabular Negative ;
end;

registration
let M be ( ( tabular Nonpositive ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Nonpositive ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
cluster M : ( ( tabular Nonpositive ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Nonpositive ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) @ : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) -> tabular Nonpositive ;
end;

registration
let M be ( ( tabular Nonnegative ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Nonnegative ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;
cluster M : ( ( tabular Nonnegative ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Nonnegative ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) @ : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) -> tabular Nonnegative ;
end;

registration
let n be ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ;
cluster (n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ,n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ) --> 1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) : ( ( V16() V21() FinSequence-like tabular ) ( V16() V21() FinSequence-like tabular ) set ) -> Positive for ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
cluster (n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ,n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ) --> (- 1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non positive V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( V16() V21() FinSequence-like tabular ) ( V16() V21() FinSequence-like tabular ) set ) -> Negative for ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
cluster (n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ,n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ) --> (- 1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non positive V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( V16() V21() FinSequence-like tabular ) ( V16() V21() FinSequence-like tabular ) set ) -> Nonpositive for ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
cluster (n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ,n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ) --> 1 : ( ( ) ( ext-real positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) : ( ( V16() V21() FinSequence-like tabular ) ( V16() V21() FinSequence-like tabular ) set ) -> Nonnegative for ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
end;

registration
let n be ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ;
cluster V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Positive Nonnegative for ( ( ) ( ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ,n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
cluster V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular Negative Nonpositive for ( ( ) ( ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) ,n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) set ) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;
end;

theorem :: MATRIX10:1
for x1 being ( ( ) ( ext-real V14() V15() ) Element of ( ( ) ( V35() V36() V37() ) set ) )
for x2 being ( ( ) ( ext-real V14() V15() ) Real) st x1 : ( ( ) ( ext-real V14() V15() ) Element of ( ( ) ( V35() V36() V37() ) set ) ) = x2 : ( ( ) ( ext-real V14() V15() ) Real) holds
- x1 : ( ( ) ( ext-real V14() V15() ) Element of ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ext-real V14() V15() ) Element of the U1 of F_Real : ( ( V118() ) ( V69() V73() V95() V115() V118() V123() V125() V127() V130() V131() V132() right-distributive left-distributive right_unital well-unital V144() left_unital ) L11()) : ( ( ) ( V35() V36() V37() ) set ) ) = - x2 : ( ( ) ( ext-real V14() V15() ) Real) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:2
for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) st [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( ) ( ) set ) holds
(- M : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ) : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) = - (M : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) )) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:3
for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) st len M1 : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( ) ( ext-real non negative V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) = len M2 : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( ) ( ext-real non negative V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & width M1 : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( ) ( ext-real non negative V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) = width M2 : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( ) ( ext-real non negative V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M1 : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( ) ( ) set ) holds
(M1 : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) - M2 : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ) : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) = (M1 : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) )) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - (M2 : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) )) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:4
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for i, j being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) st [i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ] : ( ( ) ( ) set ) in Indices M : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( ) ( ) set ) holds
(a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ) : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) = a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * (M : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) * (i : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) ,j : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) )) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:5
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds Indices M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) set ) = Indices |:M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( ) ( ) set ) ;

theorem :: MATRIX10:6
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds |:(a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) = (abs a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ) : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * |:M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;

theorem :: MATRIX10:7
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
- M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive ;

theorem :: MATRIX10:8
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive ;

theorem :: MATRIX10:9
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M2, M1 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive ;

theorem :: MATRIX10:10
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive ;

theorem :: MATRIX10:11
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative & |:M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) is_less_than |:M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive ;

theorem :: MATRIX10:12
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive ;

theorem :: MATRIX10:13
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M2, M1 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive ;

theorem :: MATRIX10:14
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) > 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive ;

theorem :: MATRIX10:15
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) < 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive ;

theorem :: MATRIX10:16
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
- M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative ;

theorem :: MATRIX10:17
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative ;

theorem :: MATRIX10:18
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative ;

theorem :: MATRIX10:19
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative & |:M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) is_less_than |:M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative ;

theorem :: MATRIX10:20
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative ;

theorem :: MATRIX10:21
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative ;

theorem :: MATRIX10:22
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) < 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative ;

theorem :: MATRIX10:23
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) > 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative ;

theorem :: MATRIX10:24
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative holds
- M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive ;

theorem :: MATRIX10:25
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive ;

theorem :: MATRIX10:26
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive ;

theorem :: MATRIX10:27
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive ;

theorem :: MATRIX10:28
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive ;

theorem :: MATRIX10:29
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive ;

theorem :: MATRIX10:30
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive ;

theorem :: MATRIX10:31
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive ;

theorem :: MATRIX10:32
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive ;

theorem :: MATRIX10:33
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds |:M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) is Nonnegative ;

theorem :: MATRIX10:34
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:35
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive holds
- M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:36
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:37
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:38
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M2, M1 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:39
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:40
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:41
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:42
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:43
for a, b being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative holds
(a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + (b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

begin

theorem :: MATRIX10:44
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:45
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:46
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3, M4 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:47
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:48
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:49
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds |:(M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) is_less_or_equal_with |:M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) + |:M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) :| : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) : ( ( tabular ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) FinSequence of K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) ;

theorem :: MATRIX10:50
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:51
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M3, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:52
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:53
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:54
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3, M4 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:55
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3, M4 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:56
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3, M4 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:57
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3, M4 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:58
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3, M4 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:59
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:60
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3, M4 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:61
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3, M4 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:62
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3, M4 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M4 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:63
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
- M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:64
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:65
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M2, M1 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st - M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
- M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:66
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:67
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:68
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:69
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive holds
M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:70
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M3, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive & M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:71
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M3, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive & M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:72
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M3, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative & M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:73
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:74
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:75
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) + M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:76
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:77
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:78
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:79
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:80
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative ;

theorem :: MATRIX10:81
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:82
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:83
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:84
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2, M3 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M3 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) - M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b1 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:85
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) > 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:86
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:87
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) < 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:88
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:89
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:90
for a being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b2 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:91
for a, b being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:92
for a, b being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonpositive & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:93
for a, b being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) < 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Negative & M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:94
for a, b being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) < b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Nonnegative & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:95
for a, b being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) >= 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) < b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_or_equal_with M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;

theorem :: MATRIX10:96
for a, b being ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )
for n being ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat)
for M1, M2 being ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of n : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) st a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) > 0 : ( ( ) ( ext-real non positive non negative V4() V9() V13() V14() V15() V33() V34() V35() V36() V37() V38() V39() V40() V41() ) Element of NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) <= b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is Positive & M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) holds
a : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M1 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) is_less_than b : ( ( ) ( ext-real V14() V15() ) Element of REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) * M2 : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( V16() V19( NAT : ( ( ) ( V35() V36() V37() V38() V39() V40() V41() ) Element of bool REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) : ( ( ) ( ) set ) ) ) V20(K291(REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) : ( ( ) ( ) M10( REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) )) ) V21() FinSequence-like tabular ) Matrix of b3 : ( ( V13() ) ( ext-real non negative V9() V13() V14() V15() ) Nat) , REAL : ( ( ) ( V4() V35() V36() V37() V41() V56() ) set ) ) ;