let n be Nat; :: thesis: for f being FinSequence of (TOP-REAL 2)
for x being set st n <= len f & x in L~ (f /^ n) holds
ex i being Nat st
( n + 1 <= i & i + 1 <= len f & x in LSeg (f,i) )
let f be FinSequence of (TOP-REAL 2); :: thesis: for x being set st n <= len f & x in L~ (f /^ n) holds
ex i being Nat st
( n + 1 <= i & i + 1 <= len f & x in LSeg (f,i) )
let x be set ; :: thesis: ( n <= len f & x in L~ (f /^ n) implies ex i being Nat st
( n + 1 <= i & i + 1 <= len f & x in LSeg (f,i) ) )
assume that
A1: n <= len f and
A2: x in L~ (f /^ n) ; :: thesis: ex i being Nat st
( n + 1 <= i & i + 1 <= len f & x in LSeg (f,i) )
consider j being Nat such that
A3: 1 <= j and
A4: j + 1 <= len (f /^ n) and
A5: x in LSeg ((f /^ n),j) by A2, SPPOL_2:13;
j + 1 <= (len f) - n by A1, A4, RFINSEQ:def 1;
then A6: n + (j + 1) <= len f by XREAL_1:19;
take n + j ; :: thesis: ( n + 1 <= n + j & (n + j) + 1 <= len f & x in LSeg (f,(n + j)) )
j + 1 <= (len f) - n by A1, A4, RFINSEQ:def 1;
hence ( n + 1 <= n + j & (n + j) + 1 <= len f & x in LSeg (f,(n + j)) ) by A3, A5, A6, SPPOL_2:5, XREAL_1:6; :: thesis: verum