let i, j be Nat; :: thesis: for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f holds
(mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j
let D be non empty set ; :: thesis: for f being FinSequence of D st i in dom f & j in dom f holds
(mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j
let f be FinSequence of D; :: thesis: ( i in dom f & j in dom f implies (mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j )
assume A1: i in dom f ; :: thesis: ( not j in dom f or (mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j )
then A2: ( 1 <= i & i <= len f ) by FINSEQ_3:25;
assume A3: j in dom f ; :: thesis: (mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j
then A4: ( 1 <= j & j <= len f ) by FINSEQ_3:25;
not mid (f,i,j) is empty by A1, A3, Th7;
then len (mid (f,i,j)) in dom (mid (f,i,j)) by FINSEQ_5:6;
hence (mid (f,i,j)) /. (len (mid (f,i,j))) = (mid (f,i,j)) . (len (mid (f,i,j))) by PARTFUN1:def 6
.= f . j by A2, A4, FINSEQ_6:189
.= f /. j by A3, PARTFUN1:def 6 ;
:: thesis: verum