let f be S-Sequence_in_R2; :: thesis:
let p, q be Point of (TOP-REAL 2); :: thesis:
let j be Nat; :: thesis:
assume that
A1: 1 <= j and
A2: j < len f and
A3: p in LSeg (f,j) and
A4: q in LSeg (p,(f /. (j + 1))) ; :: thesis:
A5: j + 1 <= len f by A2, NAT_1:13;
then A6: LSeg (f,j) = LSeg ((f /. j),(f /. (j + 1))) by A1, TOPREAL1:def 3;
A7: f /. (j + 1) in LSeg (f,j) by A1, A5, TOPREAL1:21;
then A8: LSeg (p,(f /. (j + 1))) c= LSeg (f,j) by A3, A6, TOPREAL1:6;
then A9: q in LSeg (f,j) by A4;
A10: LSeg (f,j) c= L~ f by TOPREAL3:19;

suppose p = q ; :: thesis:

hence LE p,q, L~ f,f /. 1,f /. (len f) by A3, A10, JORDAN5C:9; :: thesis:

end;

suppose A11: q <> p ; :: thesis:

for i, j being Nat st p in LSeg (f,i) & q in LSeg (f,j) & 1 <= i & i + 1 <= len f & 1 <= j & j + 1 <= len f holds
( i <= j & ( i = j implies LE p,q,f /. i,f /. (i + 1) ) )

1 <= j + 1 by NAT_1:11;
then A12: j + 1 in dom f by A5, FINSEQ_3:25;
let i1, i2 be Nat; :: thesis:
assume that
A13: p in LSeg (f,i1) and
A14: q in LSeg (f,i2) and
1 <= i1 and
A15: i1 + 1 <= len f and
A16: 1 <= i2 and
i2 + 1 <= len f ; :: thesis:
A17: p in (LSeg (f,i1)) /\ (LSeg (f,j)) by A3, A13, XBOOLE_0:def 4;
then A18: LSeg (f,i1) meets LSeg (f,j) by XBOOLE_0:4;
then A19: i1 + 1 >= j by TOPREAL1:def 7;

A20: now :: thesis:

A21: j + (1 + 1) = (j + 1) + 1 ;
assume j + 1 = i1 ; :: thesis:
then p in {(f /. (j + 1))} by A1, A15, A17, A21, TOPREAL1:def 6;
then p = f /. (j + 1) by TARSKI:def 1;
then q in {p} by A4, RLTOPSP1:70;
hence contradiction by A11, TARSKI:def 1; :: thesis:

end;

A22: now :: thesis:

assume that
A23: i2 + 1 > j and
A24: j + 1 > i2 ; :: thesis:
A25: j <= i2 by A23, NAT_1:13;
i2 <= j by A24, NAT_1:13;
hence i2 = j by A25, XXREAL_0:1; :: thesis:

end;

A26: now :: thesis:

assume that
A27: i1 + 1 > j and
A28: j + 1 > i1 ; :: thesis:
A29: j <= i1 by A27, NAT_1:13;
i1 <= j by A28, NAT_1:13;
hence i1 = j by A29, XXREAL_0:1; :: thesis:

end;

A30: q in (LSeg (f,i2)) /\ (LSeg (f,j)) by A4, A8, A14, XBOOLE_0:def 4;
then A31: LSeg (f,i2) meets LSeg (f,j) by XBOOLE_0:4;
then A32: j + 1 >= i2 by TOPREAL1:def 7;
A33: j in dom f by A1, A2, FINSEQ_3:25;

A34: now :: thesis:

assume f /. (j + 1) = f /. j ; :: thesis:
then j = j + 1 by A12, A33, PARTFUN2:10;
hence contradiction ; :: thesis:

end;

A35: now :: thesis:

A36: i2 + (1 + 1) = (i2 + 1) + 1 ;
assume i2 + 1 = j ; :: thesis:
then q in {(f /. j)} by A5, A16, A30, A36, TOPREAL1:def 6;
then q = f /. j by TARSKI:def 1;
hence contradiction by A3, A4, A6, A7, A11, A34, SPPOL_1:7, TOPREAL1:6; :: thesis:

end;

i2 + 1 >= j by A31, TOPREAL1:def 7;
then ( i2 + 1 = j or i2 = j or j + 1 = i2 ) by A22, A32, XXREAL_0:1;
then A37: i2 >= j by A35, NAT_1:11;
A38: j + 1 >= i1 by A18, TOPREAL1:def 7;
then ( i1 + 1 = j or i1 = j ) by A26, A19, A20, XXREAL_0:1;
then i1 <= j by NAT_1:11;
hence i1 <= i2 by A37, XXREAL_0:2; :: thesis:
assume A39: i1 = i2 ; :: thesis:
not p in LSeg (q,(f /. (j + 1))) by A4, A11, SPRECT_3:6;
then not LE q,p,f /. j,f /. (j + 1) by JORDAN3:30;
then LT p,q,f /. j,f /. (j + 1) by A3, A6, A14, A26, A19, A38, A20, A34, A35, A39, JORDAN3:28, XXREAL_0:1;
hence LE p,q,f /. i1,f /. (i1 + 1) by A26, A19, A38, A20, A35, A39, JORDAN3:def 6, XXREAL_0:1; :: thesis:

end;

hence LE p,q, L~ f,f /. 1,f /. (len f) by A3, A9, A10, A11, JORDAN5C:30; :: thesis:

end;

end;