:: MATRIXR1 semantic presentation

begin

theorem :: MATRIXR1:1
for r1, r2 being ( ( ) ( V11() real ext-real ) Real)
for fr1, fr2 being ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) st r1 : ( ( ) ( V11() real ext-real ) Real) = fr1 : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) & r2 : ( ( ) ( V11() real ext-real ) Real) = fr2 : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) holds
r1 : ( ( ) ( V11() real ext-real ) Real) + r2 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = fr1 : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) + fr2 : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ;

theorem :: MATRIXR1:2
for r1, r2 being ( ( ) ( V11() real ext-real ) Real)
for fr1, fr2 being ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) st r1 : ( ( ) ( V11() real ext-real ) Real) = fr1 : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) & r2 : ( ( ) ( V11() real ext-real ) Real) = fr2 : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) holds
r1 : ( ( ) ( V11() real ext-real ) Real) * r2 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = fr1 : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) * fr2 : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ;

theorem :: MATRIXR1:3
for F being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) holds
( F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + (- F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = 0* (len F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of REAL (len b1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) & F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = 0* (len F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of REAL (len b1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) ;

theorem :: MATRIXR1:4
for F1, F2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + (- F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:5
for F being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) holds F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - (0* (len F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of REAL (len b1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:6
for F being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) holds (0* (len F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of REAL (len b1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) - F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = - F : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:7
for F1, F2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - (- F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:8
for F1, F2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
- (F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:9
for F1, F2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
- (F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (- F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:10
for F1, F2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = 0* (len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of REAL (len b1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds
F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:11
for F1, F2, F3 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F3 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F3 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - (F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + F3 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:12
for F1, F2, F3 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F3 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + (F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F3 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F3 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:13
for F1, F2, F3 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F3 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - (F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F3 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + F3 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:14
for F1, F2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:15
for F1, F2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (F1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) - F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + F2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

begin

theorem :: MATRIXR1:16
for K being ( ( non empty ) ( non empty ) multMagma )
for p being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( V1() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( V1() ) set ) )
for a being ( ( ) ( ) Element of ( ( ) ( V1() ) set ) ) holds len (a : ( ( ) ( ) Element of ( ( ) ( V1() ) set ) ) * p : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( V1() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( V1() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len p : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( V1() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( V1() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: MATRIXR1:17
for r being ( ( ) ( V11() real ext-real ) Real)
for fr being ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) )
for p being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for fp being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) st r : ( ( ) ( V11() real ext-real ) Real) = fr : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) & p : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = fp : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) * p : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = fr : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) * fp : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ;

theorem :: MATRIXR1:18
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) holds Indices (a : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) : ( ( ) ( ) set ) = Indices A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ) set ) ;

theorem :: MATRIXR1:19
for i being ( ( natural ) ( ordinal natural ext-real non negative ) Nat)
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) )
for M being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) st 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) & i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) <= width M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
Col ((a : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) Function-like V32() V39( len (b3 : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * b4 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (len (b3 : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * b4 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) = a : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * (Col (M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) Function-like V32() V39( len b4 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (len b4 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) Function-like V32() V39( len b4 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (len b4 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) ;

theorem :: MATRIXR1:20
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) )
for M being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of )
for i being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) & i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) <= len M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
Line ((a : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) Function-like V32() V39( width (b2 : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * b3 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (width (b2 : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * b3 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) = a : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * (Line (M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) Function-like V32() V39( width b3 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (width b3 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) Function-like V32() V39( width b3 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (width b3 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) ;

theorem :: MATRIXR1:21
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field)
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) st width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex C being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) st
( len C : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width C : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for i, j being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st [i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ] : ( ( ) ( ) set ) in Indices C : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ) set ) holds
C : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) * (i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) = (Line (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) Function-like V32() V39( width b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (width b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) "*" (Col (B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) Function-like V32() V39( len b3 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of (len b3 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) ) ;

theorem :: MATRIXR1:22
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field)
for a being ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) )
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) st width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) * (a : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) = a : ( ( ) ( ) Element of ( ( ) ( V1() V2() ) set ) ) * (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) * B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) ;

definition
let A be ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
func MXR2MXF A -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) equals :: MATRIXR1:def 1
A : ( ( ) ( ) doubleLoopStr ) ;
end;

definition
let A be ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ;
func MXF2MXR A -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) equals :: MATRIXR1:def 2
A : ( ( ) ( ) doubleLoopStr ) ;
end;

theorem :: MATRIXR1:23
for D1, D2 being ( ( ) ( ) set )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(b1 : ( ( ) ( ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b1 : ( ( ) ( ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of D1 : ( ( ) ( ) set ) ) )
for B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(b2 : ( ( ) ( ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b2 : ( ( ) ( ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of D2 : ( ( ) ( ) set ) ) ) st A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(b1 : ( ( ) ( ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b1 : ( ( ) ( ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b1 : ( ( ) ( ) set ) ) ) = B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(b2 : ( ( ) ( ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b2 : ( ( ) ( ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b2 : ( ( ) ( ) set ) ) ) holds
for i, j being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st [i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ] : ( ( ) ( ) set ) in Indices A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(b1 : ( ( ) ( ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b1 : ( ( ) ( ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) holds
A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(b1 : ( ( ) ( ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b1 : ( ( ) ( ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b1 : ( ( ) ( ) set ) ) ) * (i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) = B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(b2 : ( ( ) ( ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b2 : ( ( ) ( ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of b2 : ( ( ) ( ) set ) ) ) * (i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( ) Element of b2 : ( ( ) ( ) set ) ) ;

theorem :: MATRIXR1:24
for K being ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field)
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) holds Indices (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) : ( ( ) ( ) set ) = Indices A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of b1 : ( ( non empty non degenerated right_complementable almost_left_invertible V98() V99() V100() V191() V193() well-unital V199() ) ( non empty non degenerated V49() right_complementable almost_left_invertible V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) Field) : ( ( ) ( V1() V2() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) : ( ( ) ( ) set ) ;

definition
let A, B be ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
func A + B -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) equals :: MATRIXR1:def 3
 MXF2MXR ((MXR2MXF A : ( ( ) ( ) doubleLoopStr ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) + (MXR2MXF B : ( ( ) ( ) VectSpStr over A : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
end;

theorem :: MATRIXR1:25
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds
( len (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for i, j being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st [i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ] : ( ( ) ( ) set ) in Indices A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ) set ) holds
(A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + (B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) ;

theorem :: MATRIXR1:26
for A, B, C being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len C : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width C : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for i, j being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st [i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ] : ( ( ) ( ) set ) in Indices A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ) set ) holds
C : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + (B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds
C : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

definition
let A be ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
func - A -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) equals :: MATRIXR1:def 4
 MXF2MXR (- (MXR2MXF A : ( ( ) ( ) doubleLoopStr ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
end;

definition
let A, B be ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
func A - B -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) equals :: MATRIXR1:def 5
 MXF2MXR ((MXR2MXF A : ( ( ) ( ) doubleLoopStr ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) - (MXR2MXF B : ( ( ) ( ) VectSpStr over A : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
func A * B -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) equals :: MATRIXR1:def 6
 MXF2MXR ((MXR2MXF A : ( ( ) ( ) doubleLoopStr ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) * (MXR2MXF B : ( ( ) ( ) VectSpStr over A : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
end;

definition
let a be ( ( real ) ( real ) number ) ;
let A be ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
func a * A -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) means :: MATRIXR1:def 7
 for ea being ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) st ea : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) = a : ( ( ) ( ) doubleLoopStr ) holds
it : ( ( Function-like V27(K7(a : ( ( ) ( ) doubleLoopStr ) ,a : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ) set ) ,a : ( ( ) ( ) doubleLoopStr ) ) ) ( V13() V16(K7(a : ( ( ) ( ) doubleLoopStr ) ,a : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ) set ) ) V17(a : ( ( ) ( ) doubleLoopStr ) ) Function-like V27(K7(a : ( ( ) ( ) doubleLoopStr ) ,a : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ) set ) ,a : ( ( ) ( ) doubleLoopStr ) ) ) Element of K6(K7(K7(a : ( ( ) ( ) doubleLoopStr ) ,a : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ) set ) ,a : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = MXF2MXR (ea : ( ( ) ( V11() real ext-real ) Element of ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) * (MXR2MXF A : ( ( ) ( ) VectSpStr over a : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
end;

theorem :: MATRIXR1:27
for a being ( ( ) ( V11() real ext-real ) Real)
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds
( len (a : ( ( ) ( V11() real ext-real ) Real) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width (a : ( ( ) ( V11() real ext-real ) Real) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: MATRIXR1:28
for a being ( ( ) ( V11() real ext-real ) Real)
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds Indices (a : ( ( ) ( V11() real ext-real ) Real) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ) set ) = Indices A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: MATRIXR1:29
for a being ( ( ) ( V11() real ext-real ) Real)
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) )
for i2, j2 being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st [i2 : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j2 : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ] : ( ( ) ( ) set ) in Indices A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ) set ) holds
(a : ( ( ) ( V11() real ext-real ) Real) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (i2 : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j2 : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = a : ( ( ) ( V11() real ext-real ) Real) * (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (i2 : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,j2 : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:30
for a being ( ( ) ( V11() real ext-real ) Real)
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( ) ( V11() real ext-real ) Real) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) @ : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = a : ( ( ) ( V11() real ext-real ) Real) * (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) @) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:31
for a being ( ( ) ( V11() real ext-real ) Real)
for i being ( ( natural ) ( ordinal natural ext-real non negative ) Nat)
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) in dom A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V145() V146() V147() V148() V149() V150() ) Element of K6(NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
( ex p being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st p : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) . i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) : ( ( ) ( FinSequence-like ) set ) & ( for q being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st q : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) . i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) : ( ( ) ( FinSequence-like ) set ) holds
(a : ( ( ) ( V11() real ext-real ) Real) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) . i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) : ( ( ) ( FinSequence-like ) set ) = a : ( ( ) ( V11() real ext-real ) Real) * q : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) ;

theorem :: MATRIXR1:32
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:33
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = 2 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:34
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = 3 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

definition
let n, m be ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ;
func 0_Rmatrix (n,m) -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) equals :: MATRIXR1:def 8
 MXF2MXR (0. (F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) ,n : ( ( ) ( ) doubleLoopStr ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) doubleLoopStr ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of n : ( ( ) ( ) doubleLoopStr ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) doubleLoopStr ) ) , the carrier of F_Real : ( ( strict ) ( non empty non degenerated V49() right_complementable almost_left_invertible strict V98() V99() V100() unital V191() V193() right-distributive left-distributive right_unital well-unital V199() left_unital ) doubleLoopStr ) : ( ( ) ( V1() V2() V145() V146() V147() ) set ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
end;

theorem :: MATRIXR1:35
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) - (- B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:36
for n, m being ( ( natural ) ( ordinal natural ext-real non negative ) Nat)
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = m : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) & n : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + (0_Rmatrix (n : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,m : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) & (0_Rmatrix (n : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ,m : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) ;

theorem :: MATRIXR1:37
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds
B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = 0_Rmatrix ((len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:38
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = 0_Rmatrix ((len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds
B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = - A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:39
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) - A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds
A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = 0_Rmatrix ((len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:40
for a being ( ( ) ( V11() real ext-real ) Real)
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (a : ( ( ) ( V11() real ext-real ) Real) * B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = a : ( ( ) ( V11() real ext-real ) Real) * (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:41
for a being ( ( ) ( V11() real ext-real ) Real)
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( ) ( V11() real ext-real ) Real) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = a : ( ( ) ( V11() real ext-real ) Real) * (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:42
for M being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + (0_Rmatrix ((len M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:43
for a being ( ( real ) ( real ) number )
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
a : ( ( real ) ( real ) number ) * (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = (a : ( ( real ) ( real ) number ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + (a : ( ( real ) ( real ) number ) * B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:44
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = 0_Rmatrix ((len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

definition
let x be ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;
assume len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ;
func ColVec2Mx x -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) means :: MATRIXR1:def 9
( len it : ( ( ) ( ) VectSpStr over x : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( ) doubleLoopStr ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width it : ( ( ) ( ) VectSpStr over x : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for j being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) in dom x : ( ( ) ( ) doubleLoopStr ) : ( ( ) ( V145() V146() V147() V148() V149() V150() ) Element of K6(NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
it : ( ( ) ( ) VectSpStr over x : ( ( ) ( ) doubleLoopStr ) ) . j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) : ( ( ) ( ) set ) = <*(x : ( ( ) ( ) doubleLoopStr ) . j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) *> : ( ( ) ( V1() V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39(1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) );
end;

theorem :: MATRIXR1:45
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for M being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = ColVec2Mx x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) iff ( Col (M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( len b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (len b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) & width M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: MATRIXR1:46
for x1, x2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ColVec2Mx (x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = (ColVec2Mx x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + (ColVec2Mx x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:47
for a being ( ( ) ( V11() real ext-real ) Real)
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ColVec2Mx (a : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = a : ( ( ) ( V11() real ext-real ) Real) * (ColVec2Mx x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

definition
let x be ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;
func LineVec2Mx x -> ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) means :: MATRIXR1:def 10
( width it : ( ( ) ( ) VectSpStr over x : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( ) doubleLoopStr ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len it : ( ( ) ( ) VectSpStr over x : ( ( ) ( ) doubleLoopStr ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for j being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) in dom x : ( ( ) ( ) doubleLoopStr ) : ( ( ) ( V145() V146() V147() V148() V149() V150() ) Element of K6(NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
it : ( ( ) ( ) VectSpStr over x : ( ( ) ( ) doubleLoopStr ) ) * (1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = x : ( ( ) ( ) doubleLoopStr ) . j : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) );
end;

theorem :: MATRIXR1:48
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for M being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) holds
( M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = LineVec2Mx x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) iff ( Line (M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( width b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (width b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) & len M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: MATRIXR1:49
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (LineVec2Mx x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) @ : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = ColVec2Mx x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) & (ColVec2Mx x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) @ : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = LineVec2Mx x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) ;

theorem :: MATRIXR1:50
for x1, x2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LineVec2Mx (x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = (LineVec2Mx x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + (LineVec2Mx x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:51
for a being ( ( ) ( V11() real ext-real ) Real)
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) holds LineVec2Mx (a : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = a : ( ( ) ( V11() real ext-real ) Real) * (LineVec2Mx x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

definition
let M be ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
let x be ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;
func M * x -> ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) equals :: MATRIXR1:def 11
 Col ((M : ( ( ) ( ) doubleLoopStr ) * (ColVec2Mx x : ( ( ) ( ) VectSpStr over M : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( len (M : ( ( ) ( ) doubleLoopStr ) * (ColVec2Mx x : ( ( ) ( ) VectSpStr over M : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (len (M : ( ( ) ( ) doubleLoopStr ) * (ColVec2Mx x : ( ( ) ( ) VectSpStr over M : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
func x * M -> ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) equals :: MATRIXR1:def 12
 Line (((LineVec2Mx x : ( ( ) ( ) VectSpStr over M : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * M : ( ( ) ( ) doubleLoopStr ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( width ((LineVec2Mx x : ( ( ) ( ) VectSpStr over M : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * M : ( ( ) ( ) doubleLoopStr ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (width ((LineVec2Mx x : ( ( ) ( ) VectSpStr over M : ( ( ) ( ) doubleLoopStr ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * M : ( ( ) ( ) doubleLoopStr ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;
end;

theorem :: MATRIXR1:52
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & ( len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) or width (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) @) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) @) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:53
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & ( width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) or len (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) @) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) @) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:54
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for i being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) & i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) <= width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
Col ((A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( len (b1 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (len (b1 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = (Col (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( len b1 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (len b1 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + (Col (B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( len b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (len b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:55
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for i being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) & i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) <= len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
Line ((A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( width (b1 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (width (b1 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = (Line (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( width b1 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (width b1 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + (Line (B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( width b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (width b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:56
for a being ( ( ) ( V11() real ext-real ) Real)
for M being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) )
for i being ( ( natural ) ( ordinal natural ext-real non negative ) Nat) st 1 : ( ( ) ( V1() ordinal natural V11() real V30() V31() ext-real positive non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) & i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) <= width M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
Col ((a : ( ( ) ( V11() real ext-real ) Real) * M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( len (b1 : ( ( ) ( V11() real ext-real ) Real) * b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (len (b1 : ( ( ) ( V11() real ext-real ) Real) * b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) = a : ( ( ) ( V11() real ext-real ) Real) * (Col (M : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ,i : ( ( natural ) ( ordinal natural ext-real non negative ) Nat) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( len b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (len b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() V39( len b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of (len b2 : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) -tuples_on REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:57
for x1, x2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:58
for x1, x2 being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (x1 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + (x2 : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:59
for a being ( ( ) ( V11() real ext-real ) Real)
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * (a : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = a : ( ( ) ( V11() real ext-real ) Real) * (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:60
for a being ( ( ) ( V11() real ext-real ) Real)
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = a : ( ( ) ( V11() real ext-real ) Real) * (x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:61
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
len (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: MATRIXR1:62
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
len (x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: MATRIXR1:63
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + (B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:64
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) )
for A, B being ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) st len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & width A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = width B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * (A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) + B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = (x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * A : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) + (x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * B : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ;

theorem :: MATRIXR1:65
for n, m being ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = m : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(0_Rmatrix (n : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) * x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = 0* n : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;

theorem :: MATRIXR1:66
for n, m being ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) st len x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real functional V30() V31() FinSequence-membered ext-real non positive non negative V145() V146() V147() V148() V149() V150() V151() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) holds
x : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) * (0_Rmatrix (n : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) * : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like tabular ) Matrix of ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) FinSequence of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) = 0* m : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-yielding V109() V110() ) Element of REAL b2 : ( ( ) ( ordinal natural V11() real V30() V31() ext-real non negative V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( V145() V146() V147() V148() V149() V150() V151() ) Element of K6(REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V32() V145() V146() V147() V151() ) set ) ) ) ;