let f be S-Sequence_in_R2; :: thesis: for Q being closed Subset of (TOP-REAL 2) st L~ f meets Q & not f /. (len f) in Q holds
(L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q))) /\ Q = {(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)}
let Q be closed Subset of (TOP-REAL 2); :: thesis: ( L~ f meets Q & not f /. (len f) in Q implies (L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q))) /\ Q = {(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)} )
assume that
A1: L~ f meets Q and
A2: not f /. (len f) in Q ; :: thesis: (L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q))) /\ Q = {(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)}
set p1 = f /. 1;
set p2 = f /. (len f);
set lp = Last_Point (L~ f),(f /. 1),(f /. (len f)),Q;
A3: (L~ f) /\ Q is closed by TOPS_1:35;
len f >= 1 + 1 by TOPREAL1:def 10;
then A4: len f > 1 by NAT_1:13;
L~ f is_an_arc_of f /. 1,f /. (len f) by TOPREAL1:31;
then A5: Last_Point (L~ f),(f /. 1),(f /. (len f)),Q in (L~ f) /\ Q by A1, A3, JORDAN5C:def 2;
then A6: Last_Point (L~ f),(f /. 1),(f /. (len f)),Q in L~ f by XBOOLE_0:def 4;
then A7: 1 <= Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f by JORDAN3:41;

A8: now

set m = mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f);
assume not (L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q))) /\ Q c= {(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)} ; :: thesis: contradiction
then consider q being set such that
A9: q in (L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q))) /\ Q and
A10: not q in {(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)} by TARSKI:def 3;
reconsider q = q as Point of (TOP-REAL 2) by A9;
A11: q in L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)) by A9, XBOOLE_0:def 4;
A12: L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)) c= L~ f by A6, JORDAN3:77;
A13: Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f < len f by A6, JORDAN3:41;
then A14: (Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1 <= len f by NAT_1:13;
A15: 1 <= (Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1 by NAT_1:11;
then len (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) = ((len f) -' ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1)) + 1 by A14, JORDAN4:20;
then A16: not mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f) is empty by CARD_1:47, NAT_1:11;
A17: q <> Last_Point (L~ f),(f /. 1),(f /. (len f)),Q by A10, TARSKI:def 1;
q in Q by A9, XBOOLE_0:def 4;
then A18: LE q, Last_Point (L~ f),(f /. 1),(f /. (len f)),Q, L~ f,f /. 1,f /. (len f) by A3, A11, A12, JORDAN5C:16;

per cases ( Last_Point (L~ f),(f /. 1),(f /. (len f)),Q = f . ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1) or Last_Point (L~ f),(f /. 1),(f /. (len f)),Q <> f . ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1) ) ;

suppose Last_Point (L~ f),(f /. 1),(f /. (len f)),Q = f . ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1) ; :: thesis: contradiction

then A19: q in L~ (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) by A11, JORDAN3:def 4;

now

assume (Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1 >= len f ; :: thesis: contradiction
then (Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1 = len f by A14, XXREAL_0:1;
then len (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) = 1 by A4, JORDAN4:27;
hence contradiction by A19, TOPREAL1:28; :: thesis: verum

end;

then A20: LE f /. ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),q, L~ f,f /. 1,f /. (len f) by A19, Th4, NAT_1:11;
A21: f /. ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1) in LSeg (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),(f /. ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1)) by RLTOPSP1:69;
Last_Point (L~ f),(f /. 1),(f /. (len f)),Q in LSeg f,(Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) by A6, JORDAN3:42;
then LE Last_Point (L~ f),(f /. 1),(f /. (len f)),Q,f /. ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1), L~ f,f /. 1,f /. (len f) by A7, A13, A21, Th5;
then LE Last_Point (L~ f),(f /. 1),(f /. (len f)),Q,q, L~ f,f /. 1,f /. (len f) by A20, JORDAN5C:13;
hence contradiction by A17, A18, JORDAN5C:12, TOPREAL1:31; :: thesis: verum

end;

suppose A22: Last_Point (L~ f),(f /. 1),(f /. (len f)),Q <> f . ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1) ; :: thesis: contradiction

A23: Last_Point (L~ f),(f /. 1),(f /. (len f)),Q in LSeg f,(Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) by A6, JORDAN3:42;
1 <= (Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1 by NAT_1:11;
then A24: (Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1 in dom f by A14, FINSEQ_3:27;
q in L~ (<*(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)*> ^ (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f))) by A11, A22, JORDAN3:def 4;
then A25: q in (L~ (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f))) \/ (LSeg ((mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) /. 1),(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)) by A16, SPPOL_2:20;

now

per cases ( q in L~ (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) or q in LSeg (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),((mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) /. 1) ) by A25, XBOOLE_0:def 3;

suppose A26: q in L~ (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) ; :: thesis: contradiction

now

assume (Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1 >= len f ; :: thesis: contradiction
then (Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1 = len f by A14, XXREAL_0:1;
then len (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) = 1 by A4, JORDAN4:27;
hence contradiction by A26, TOPREAL1:28; :: thesis: verum

end;

then A27: LE f /. ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),q, L~ f,f /. 1,f /. (len f) by A26, Th4, NAT_1:11;
f /. ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1) in LSeg (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),(f /. ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1)) by RLTOPSP1:69;
then LE Last_Point (L~ f),(f /. 1),(f /. (len f)),Q,f /. ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1), L~ f,f /. 1,f /. (len f) by A7, A13, A23, Th5;
then LE Last_Point (L~ f),(f /. 1),(f /. (len f)),Q,q, L~ f,f /. 1,f /. (len f) by A27, JORDAN5C:13;
hence contradiction by A17, A18, JORDAN5C:12, TOPREAL1:31; :: thesis: verum

end;

suppose A28: q in LSeg (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),((mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) /. 1) ; :: thesis: contradiction

1 in dom (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) by A16, FINSEQ_5:6;
then (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) /. 1 = (mid f,((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1),(len f)) . 1 by PARTFUN1:def 8
.= f . ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1) by A4, A14, A15, FINSEQ_6:124
.= f /. ((Index (Last_Point (L~ f),(f /. 1),(f /. (len f)),Q),f) + 1) by A24, PARTFUN1:def 8 ;
then LE Last_Point (L~ f),(f /. 1),(f /. (len f)),Q,q, L~ f,f /. 1,f /. (len f) by A7, A13, A23, A28, Th5;
hence contradiction by A17, A18, JORDAN5C:12, TOPREAL1:31; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum

end;

end;

end;

A29: Last_Point (L~ f),(f /. 1),(f /. (len f)),Q in Q by A5, XBOOLE_0:def 4;
len f in dom f by A4, FINSEQ_3:27;
then Last_Point (L~ f),(f /. 1),(f /. (len f)),Q <> f . (len f) by A2, A29, PARTFUN1:def 8;
then Last_Point (L~ f),(f /. 1),(f /. (len f)),Q in L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)) by A6, JORDAN5B:22;
then Last_Point (L~ f),(f /. 1),(f /. (len f)),Q in (L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q))) /\ Q by A29, XBOOLE_0:def 4;
hence (L~ (L_Cut f,(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q))) /\ Q = {(Last_Point (L~ f),(f /. 1),(f /. (len f)),Q)} by A8, ZFMISC_1:39; :: thesis: verum