let f be FinSequence of (TOP-REAL 2); :: thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
ex i being Nat st
( 1 <= i & i + 1 <= len f & p in LSeg ((f /. i),(f /. (i + 1))) )
let p be Point of (TOP-REAL 2); :: thesis: ( p in L~ f implies ex i being Nat st
( 1 <= i & i + 1 <= len f & p in LSeg ((f /. i),(f /. (i + 1))) ) )
assume p in L~ f ; :: thesis: ex i being Nat st
( 1 <= i & i + 1 <= len f & p in LSeg ((f /. i),(f /. (i + 1))) )
then consider i being Nat such that
A1: 1 <= i and
A2: i + 1 <= len f and
A3: p in LSeg (f,i) by Th13;
take i ; :: thesis: ( 1 <= i & i + 1 <= len f & p in LSeg ((f /. i),(f /. (i + 1))) )
thus ( 1 <= i & i + 1 <= len f ) by A1, A2; :: thesis: p in LSeg ((f /. i),(f /. (i + 1)))
thus p in LSeg ((f /. i),(f /. (i + 1))) by A1, A2, A3, TOPREAL1:def 3; :: thesis: verum