let f be FinSequence of (TOP-REAL 2); :: thesis: for p being Point of (TOP-REAL 2)
for i being Nat st 1 <= i & i + 1 <= len f & p in LSeg ((f /. i),(f /. (i + 1))) holds
p in L~ f
let p be Point of (TOP-REAL 2); :: thesis: for i being Nat st 1 <= i & i + 1 <= len f & p in LSeg ((f /. i),(f /. (i + 1))) holds
p in L~ f
let i be Nat; :: thesis: ( 1 <= i & i + 1 <= len f & p in LSeg ((f /. i),(f /. (i + 1))) implies p in L~ f )
assume that
A1: 1 <= i and
A2: i + 1 <= len f and
A3: p in LSeg ((f /. i),(f /. (i + 1))) ; :: thesis: p in L~ f
set X = { (LSeg (f,j)) where j is Nat : ( 1 <= j & j + 1 <= len f ) } ;
A4: LSeg (f,i) in { (LSeg (f,j)) where j is Nat : ( 1 <= j & j + 1 <= len f ) } by A1, A2;
LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A1, A2, TOPREAL1:def 3;
hence p in L~ f by A3, A4, TARSKI:def 4; :: thesis: verum