let n be Nat; for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p, x being Point of (TOP-REAL 2) st x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = W-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); for p, x being Point of (TOP-REAL 2) st x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = W-bound (L~ (Cage (C,n)))
let p, x be Point of (TOP-REAL 2); ( x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) implies p `1 = W-bound (L~ (Cage (C,n))) )
set G = Gauge (C,n);
set f = Cage (C,n);
A1:
Cage (C,n) is_sequence_on Gauge (C,n)
by JORDAN9:def 1;
assume A2:
x in W-most C
; ( not p in (west_halfline x) /\ (L~ (Cage (C,n))) or p `1 = W-bound (L~ (Cage (C,n))) )
then A3:
x in C
by XBOOLE_0:def 4;
A4:
len (Gauge (C,n)) = width (Gauge (C,n))
by JORDAN8:def 1;
assume A5:
p in (west_halfline x) /\ (L~ (Cage (C,n)))
; p `1 = W-bound (L~ (Cage (C,n)))
then
p in L~ (Cage (C,n))
by XBOOLE_0:def 4;
then consider i being Nat such that
A6:
1 <= i
and
A7:
i + 1 <= len (Cage (C,n))
and
A8:
p in LSeg ((Cage (C,n)),i)
by SPPOL_2:13;
A9:
LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1)))
by A6, A7, TOPREAL1:def 3;
A10:
i < len (Cage (C,n))
by A7, NAT_1:13;
then
i in Seg (len (Cage (C,n)))
by A6, FINSEQ_1:1;
then
i in dom (Cage (C,n))
by FINSEQ_1:def 3;
then consider i1, i2 being Nat such that
A11:
[i1,i2] in Indices (Gauge (C,n))
and
A12:
(Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2)
by A1, GOBOARD1:def 9;
A13:
( 1 <= i2 & i2 <= width (Gauge (C,n)) )
by A11, MATRIX_0:32;
p in west_halfline x
by A5, XBOOLE_0:def 4;
then
LSeg ((Cage (C,n)),i) is vertical
by A2, A6, A8, A10, Th81;
then
((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1
by A9, SPPOL_1:16;
then A14:
p `1 = ((Cage (C,n)) /. i) `1
by A8, A9, GOBOARD7:5;
A15:
i1 <= len (Gauge (C,n))
by A11, MATRIX_0:32;
A16:
1 <= i1
by A11, MATRIX_0:32;
x `1 =
(W-min C) `1
by A2, PSCOMP_1:31
.=
W-bound C
by EUCLID:52
.=
((Gauge (C,n)) * (2,i2)) `1
by A4, A13, JORDAN8:11
;
then
i1 < 1 + 1
by A3, A5, A12, A13, A15, A14, Th77, SPRECT_3:13;
then
i1 <= 1
by NAT_1:13;
then
i1 = 1
by A16, XXREAL_0:1;
then
(Cage (C,n)) /. i in W-most (L~ (Cage (C,n)))
by A6, A10, A12, A13, Th59;
hence
p `1 = W-bound (L~ (Cage (C,n)))
by A14, Th6; verum