let f be FinSequence of (TOP-REAL 2); :: thesis: for i being Nat st 1 <= i & i + 1 <= len f & f is being_S-Seq & Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) in LSeg (f,i) holds

Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. (i + 1)

let i be Nat; :: thesis: ( 1 <= i & i + 1 <= len f & f is being_S-Seq & Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) in LSeg (f,i) implies Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. (i + 1) )

assume that

A1: ( 1 <= i & i + 1 <= len f ) and

A2: f is being_S-Seq and

A3: Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) in LSeg (f,i) ; :: thesis: Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. (i + 1)

reconsider Q = LSeg (f,i) as non empty Subset of (TOP-REAL 2) by A3;

Q = LSeg ((f /. i),(f /. (i + 1))) by A1, TOPREAL1:def 3;

then Q c= L~ f by A1, SPPOL_2:16;

then L~ f meets Q by A3, XBOOLE_0:3;

then A4: Last_Point ((L~ f),(f /. 1),(f /. (len f)),Q) = Last_Point (Q,(f /. i),(f /. (i + 1)),Q) by A1, A2, A3, Th20;

( Q is closed & Q is_an_arc_of f /. i,f /. (i + 1) ) by A1, A2, JORDAN5B:15;

hence Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. (i + 1) by A4, Th7; :: thesis: verum

Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. (i + 1)

let i be Nat; :: thesis: ( 1 <= i & i + 1 <= len f & f is being_S-Seq & Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) in LSeg (f,i) implies Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. (i + 1) )

assume that

A1: ( 1 <= i & i + 1 <= len f ) and

A2: f is being_S-Seq and

A3: Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) in LSeg (f,i) ; :: thesis: Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. (i + 1)

reconsider Q = LSeg (f,i) as non empty Subset of (TOP-REAL 2) by A3;

Q = LSeg ((f /. i),(f /. (i + 1))) by A1, TOPREAL1:def 3;

then Q c= L~ f by A1, SPPOL_2:16;

then L~ f meets Q by A3, XBOOLE_0:3;

then A4: Last_Point ((L~ f),(f /. 1),(f /. (len f)),Q) = Last_Point (Q,(f /. i),(f /. (i + 1)),Q) by A1, A2, A3, Th20;

( Q is closed & Q is_an_arc_of f /. i,f /. (i + 1) ) by A1, A2, JORDAN5B:15;

hence Last_Point ((L~ f),(f /. 1),(f /. (len f)),(LSeg (f,i))) = f /. (i + 1) by A4, Th7; :: thesis: verum