let i, j be Nat; :: thesis: for D being non empty set
for f being FinSequence of D
for M being Matrix of D st [i,j] in Indices M & M . i = f holds
( len f = width M & j in dom f )
let D be non empty set ; :: thesis: for f being FinSequence of D
for M being Matrix of D st [i,j] in Indices M & M . i = f holds
( len f = width M & j in dom f )
let f be FinSequence of D; :: thesis: for M being Matrix of D st [i,j] in Indices M & M . i = f holds
( len f = width M & j in dom f )
let M be Matrix of D; :: thesis: ( [i,j] in Indices M & M . i = f implies ( len f = width M & j in dom f ) )
assume A1: [i,j] in Indices M ; :: thesis: ( not M . i = f or ( len f = width M & j in dom f ) )
A2: j in Seg (width M) by A1, ZFMISC_1:87;
not M is empty by A1, ZFMISC_1:87;
then len M > 0 ;
then consider p being FinSequence such that
A3: p in rng M and
A4: len p = width M by Def3;
consider n being Nat such that
A5: for x being object st x in rng M holds
ex s being FinSequence st
( s = x & len s = n ) by Def1;
i in dom M by A1, ZFMISC_1:87;
then M . i in rng M by FUNCT_1:def 3;
then A6: ex s being FinSequence st
( s = M . i & len s = n ) by A5;
ex s being FinSequence st
( s = p & len s = n ) by A3, A5;
hence ( not M . i = f or ( len f = width M & j in dom f ) ) by A2, A4, A6, FINSEQ_1:def 3; :: thesis: verum