let z be constant standard clockwise_oriented special_circular_sequence; ( z /. 1 = N-min (L~ z) & E-min (L~ z) <> S-max (L~ z) implies (E-min (L~ z)) .. z < (S-max (L~ z)) .. z )
set i1 = (E-min (L~ z)) .. z;
set i2 = (S-max (L~ z)) .. z;
assume that
A1:
z /. 1 = N-min (L~ z)
and
A2:
( E-min (L~ z) <> S-max (L~ z) & (E-min (L~ z)) .. z >= (S-max (L~ z)) .. z )
; contradiction
A3:
S-bound (L~ z) < N-bound (L~ z)
by TOPREAL5:16;
z /. 2 in N-most (L~ z)
by A1, Th30;
then A4: (z /. 2) `2 =
(N-min (L~ z)) `2
by PSCOMP_1:39
.=
N-bound (L~ z)
by EUCLID:52
;
A5:
S-max (L~ z) in rng z
by Th42;
then A6:
(S-max (L~ z)) .. z in dom z
by FINSEQ_4:20;
then A7:
(S-max (L~ z)) .. z <= len z
by FINSEQ_3:25;
A8: z /. ((S-max (L~ z)) .. z) =
z . ((S-max (L~ z)) .. z)
by A6, PARTFUN1:def 6
.=
S-max (L~ z)
by A5, FINSEQ_4:19
;
then A9:
(z /. ((S-max (L~ z)) .. z)) `2 = S-bound (L~ z)
by EUCLID:52;
A10:
1 <= (S-max (L~ z)) .. z
by A6, FINSEQ_3:25;
A11:
(S-max (L~ z)) .. z <> 0
by A6, FINSEQ_3:25;
(z /. 1) `2 = N-bound (L~ z)
by A1, EUCLID:52;
then
(S-max (L~ z)) .. z <> 0 & ... & (S-max (L~ z)) .. z <> 2
by A4, A11, A9, A3;
then A12:
(S-max (L~ z)) .. z > 2
;
then reconsider h = mid (z,((S-max (L~ z)) .. z),2) as S-Sequence_in_R2 by A7, Th37;
A13:
2 <= len z
by NAT_D:60;
then A14:
2 in dom z
by FINSEQ_3:25;
then
h /. 1 = S-max (L~ z)
by A6, A8, Th8;
then A15:
(h /. 1) `2 = S-bound (L~ z)
by EUCLID:52;
( h is_in_the_area_of z & h /. (len h) = z /. 2 )
by A6, A14, Th9, Th21, Th22;
then A16:
( len h >= 2 & h is_a_v.c._for z )
by A4, A15, TOPREAL1:def 8;
N-max (L~ z) in L~ z
by SPRECT_1:11;
then
(N-max (L~ z)) `1 <= E-bound (L~ z)
by PSCOMP_1:24;
then
(N-min (L~ z)) `1 < E-bound (L~ z)
by Th51, XXREAL_0:2;
then A17:
(N-min (L~ z)) `1 < (E-min (L~ z)) `1
by EUCLID:52;
A18:
E-min (L~ z) in rng z
by Th45;
then A19:
(E-min (L~ z)) .. z in dom z
by FINSEQ_4:20;
then A20: z /. ((E-min (L~ z)) .. z) =
z . ((E-min (L~ z)) .. z)
by PARTFUN1:def 6
.=
E-min (L~ z)
by A18, FINSEQ_4:19
;
A21:
(E-min (L~ z)) .. z <= len z
by A19, FINSEQ_3:25;
z /. (len z) = N-min (L~ z)
by A1, FINSEQ_6:def 1;
then A22:
(E-min (L~ z)) .. z < len z
by A21, A20, A17, XXREAL_0:1;
then
((E-min (L~ z)) .. z) + 1 <= len z
by NAT_1:13;
then
(len z) - ((E-min (L~ z)) .. z) >= 1
by XREAL_1:19;
then
(len z) -' ((E-min (L~ z)) .. z) >= 1
by NAT_D:39;
then A23:
((len z) -' ((E-min (L~ z)) .. z)) + 1 >= 1 + 1
by XREAL_1:6;
A24:
(E-min (L~ z)) .. z > (S-max (L~ z)) .. z
by A2, A8, A20, XXREAL_0:1;
then
(E-min (L~ z)) .. z > 1
by A10, XXREAL_0:2;
then reconsider M = mid (z,(len z),((E-min (L~ z)) .. z)) as S-Sequence_in_R2 by A22, Th37;
A25:
len z in dom z
by FINSEQ_5:6;
then A26: M /. (len M) =
z /. ((E-min (L~ z)) .. z)
by A19, Th9
.=
E-min (L~ z)
by A18, FINSEQ_5:38
;
1 <= (E-min (L~ z)) .. z
by A19, FINSEQ_3:25;
then A27:
len M = ((len z) -' ((E-min (L~ z)) .. z)) + 1
by A21, FINSEQ_6:187;
A28:
L~ M misses L~ h
by A21, A24, A12, Th49;
A29:
z /. 1 = z /. (len z)
by FINSEQ_6:def 1;
then A30:
M /. 1 = z /. 1
by A19, A25, Th8;
per cases
( NW-corner (L~ z) = N-min (L~ z) or NW-corner (L~ z) <> N-min (L~ z) )
;
suppose A31:
NW-corner (L~ z) = N-min (L~ z)
;
contradiction
M /. 1
= z /. (len z)
by A19, A25, Th8;
then A32:
(M /. 1) `1 = W-bound (L~ z)
by A1, A29, A31, EUCLID:52;
(
M is_in_the_area_of z &
(M /. (len M)) `1 = E-bound (L~ z) )
by A19, A25, A26, Th21, Th22, EUCLID:52;
then
M is_a_h.c._for z
by A32;
hence
contradiction
by A16, A28, A27, A23, Th29;
verum end; suppose
NW-corner (L~ z) <> N-min (L~ z)
;
contradictionthen reconsider g =
<*(NW-corner (L~ z))*> ^ M as
S-Sequence_in_R2 by A1, A19, A25, A29, Th66;
A33:
(
len g >= 2 &
L~ g = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) )
by SPPOL_2:20, TOPREAL1:def 8;
g /. 1
= NW-corner (L~ z)
by FINSEQ_5:15;
then A34:
(g /. 1) `1 = W-bound (L~ z)
by EUCLID:52;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z)
by A10, A7, A13, JORDAN4:35, XBOOLE_1:26;
then A35:
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)}
by A1, A30, PSCOMP_1:43;
(
len M in dom M &
len g = (len M) + (len <*(NW-corner (L~ z))*>) )
by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) =
M /. (len M)
by FINSEQ_4:69
.=
z /. ((E-min (L~ z)) .. z)
by A19, A25, Th9
.=
E-min (L~ z)
by A18, FINSEQ_5:38
;
then A36:
(g /. (len g)) `1 = E-bound (L~ z)
by EUCLID:52;
A37:
M /. 1
in L~ M
by A27, A23, JORDAN3:1;
(
M is_in_the_area_of z &
<*(NW-corner (L~ z))*> is_in_the_area_of z )
by A19, A25, Th21, Th22, Th26;
then
g is_in_the_area_of z
by Th24;
then
g is_a_h.c._for z
by A34, A36;
hence
contradiction
by A16, A28, A33, A35, A37, Th29, ZFMISC_1:125;
verum end;