let f be FinSequence of (); :: thesis: for i being Nat st 1 <= i & i + 1 <= len f & f is being_S-Seq & First_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) in LSeg (f,i) holds
First_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. i

let i be Nat; :: thesis: ( 1 <= i & i + 1 <= len f & f is being_S-Seq & First_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) in LSeg (f,i) implies First_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. i )
assume that
A1: ( 1 <= i & i + 1 <= len f ) and
A2: f is being_S-Seq and
A3: First_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) in LSeg (f,i) ; :: thesis: First_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. i
reconsider Q = LSeg (f,i) as non empty Subset of () by A3;
Q = LSeg ((f /. i),(f /. (i + 1))) by ;
then Q c= L~ f by ;
then L~ f meets Q by ;
then A4: First_Point ((L~ f),(f /. 1),(f /. (len f)),Q) = First_Point (Q,(f /. i),(f /. (i + 1)),Q) by A1, A2, A3, Th19;
( Q is closed & Q is_an_arc_of f /. i,f /. (i + 1) ) by ;
hence First_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. i by ; :: thesis: verum