let z be constant standard clockwise_oriented special_circular_sequence; ( z /. 1 = N-min (L~ z) implies (N-max (L~ z)) .. z < (S-min (L~ z)) .. z )
set i1 = (N-max (L~ z)) .. z;
set i2 = (S-min (L~ z)) .. z;
assume that
A1:
z /. 1 = N-min (L~ z)
and
A2:
(N-max (L~ z)) .. z >= (S-min (L~ z)) .. z
; contradiction
A3:
N-min (L~ z) <> N-max (L~ z)
by Th52;
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-bound (L~ z) < N-bound (L~ z)
by TOPREAL5:16;
A6:
S-min (L~ z) in rng z
by Th41;
then A7:
(S-min (L~ z)) .. z in dom z
by FINSEQ_4:20;
then A8:
(S-min (L~ z)) .. z <= len z
by FINSEQ_3:25;
A9: z /. ((S-min (L~ z)) .. z) =
z . ((S-min (L~ z)) .. z)
by A7, PARTFUN1:def 6
.=
S-min (L~ z)
by A6, FINSEQ_4:19
;
then A10:
(z /. ((S-min (L~ z)) .. z)) `2 = S-bound (L~ z)
by EUCLID:52;
A11:
1 <= (S-min (L~ z)) .. z
by A7, FINSEQ_3:25;
A12:
(S-min (L~ z)) .. z <> 0
by A7, FINSEQ_3:25;
(z /. 1) `2 = N-bound (L~ z)
by A1, EUCLID:52;
then
(S-min (L~ z)) .. z <> 0 & ... & (S-min (L~ z)) .. z <> 2
by A4, A12, A10, A5;
then A13:
(S-min (L~ z)) .. z > 2
;
then reconsider h = mid (z,((S-min (L~ z)) .. z),2) as S-Sequence_in_R2 by A8, Th37;
A14:
2 <= len z
by NAT_D:60;
then A15:
2 in dom z
by FINSEQ_3:25;
then
h /. 1 = S-min (L~ z)
by A7, A9, Th8;
then A16:
(h /. 1) `2 = S-bound (L~ z)
by EUCLID:52;
( h is_in_the_area_of z & h /. (len h) = z /. 2 )
by A7, A15, Th9, Th21, Th22;
then A17:
( len h >= 2 & h is_a_v.c._for z )
by A4, A16, TOPREAL1:def 8;
A18:
N-max (L~ z) in rng z
by Th40;
then A19:
(N-max (L~ z)) .. z in dom z
by FINSEQ_4:20;
then A20: z /. ((N-max (L~ z)) .. z) =
z . ((N-max (L~ z)) .. z)
by PARTFUN1:def 6
.=
N-max (L~ z)
by A18, FINSEQ_4:19
;
A21:
(N-max (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:
(N-max (L~ z)) .. z < len z
by A21, A20, A3, XXREAL_0:1;
then
((N-max (L~ z)) .. z) + 1 <= len z
by NAT_1:13;
then
(len z) - ((N-max (L~ z)) .. z) >= 1
by XREAL_1:19;
then
(len z) -' ((N-max (L~ z)) .. z) >= 1
by NAT_D:39;
then A23:
((len z) -' ((N-max (L~ z)) .. z)) + 1 >= 1 + 1
by XREAL_1:6;
( (N-max (L~ z)) `2 = N-bound (L~ z) & (S-min (L~ z)) `2 = S-bound (L~ z) )
by EUCLID:52;
then
z /. ((N-max (L~ z)) .. z) <> z /. ((S-min (L~ z)) .. z)
by A9, A20, TOPREAL5:16;
then A24:
(N-max (L~ z)) .. z > (S-min (L~ z)) .. z
by A2, XXREAL_0:1;
then
(N-max (L~ z)) .. z > 1
by A11, XXREAL_0:2;
then reconsider M = mid (z,(len z),((N-max (L~ z)) .. z)) as S-Sequence_in_R2 by A22, Th37;
A25:
1 in dom M
by FINSEQ_5:6;
A26:
len z in dom z
by FINSEQ_5:6;
then A27: M /. (len M) =
z /. ((N-max (L~ z)) .. z)
by A19, Th9
.=
N-max (L~ z)
by A18, FINSEQ_5:38
;
A28:
L~ M misses L~ h
by A21, A24, A13, Th49;
1 <= (N-max (L~ z)) .. z
by A19, FINSEQ_3:25;
then A29:
len M = ((len z) -' ((N-max (L~ z)) .. z)) + 1
by A21, FINSEQ_6:187;
then A30:
M /. (len M) in L~ M
by A23, JORDAN3:1;
A31:
z /. 1 = z /. (len z)
by FINSEQ_6:def 1;
then A32:
M /. 1 = z /. 1
by A19, A26, Th8;
per cases
( ( NW-corner (L~ z) = N-min (L~ z) & NE-corner (L~ z) = N-max (L~ z) ) or ( NW-corner (L~ z) = N-min (L~ z) & NE-corner (L~ z) <> N-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) = N-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) <> N-max (L~ z) ) )
;
suppose that A33:
NW-corner (L~ z) = N-min (L~ z)
and A34:
NE-corner (L~ z) = N-max (L~ z)
;
contradictionA35:
(M /. (len M)) `1 = E-bound (L~ z)
by A27, A34, EUCLID:52;
M /. 1
= z /. (len z)
by A19, A26, Th8;
then A36:
(M /. 1) `1 = W-bound (L~ z)
by A1, A31, A33, EUCLID:52;
M is_in_the_area_of z
by A19, A26, Th21, Th22;
then
M is_a_h.c._for z
by A36, A35;
hence
contradiction
by A17, A28, A29, A23, Th29;
verum end; suppose that A37:
NW-corner (L~ z) = N-min (L~ z)
and A38:
NE-corner (L~ z) <> N-max (L~ z)
;
contradictionreconsider g =
M ^ <*(NE-corner (L~ z))*> as
S-Sequence_in_R2 by A19, A20, A26, A38, Th62;
A39:
(
len g >= 2 &
L~ g = (L~ M) \/ (LSeg ((M /. (len M)),(NE-corner (L~ z)))) )
by SPPOL_2:19, TOPREAL1:def 8;
len g =
(len M) + (len <*(NE-corner (L~ z))*>)
by FINSEQ_1:22
.=
(len M) + 1
by FINSEQ_1:39
;
then
g /. (len g) = NE-corner (L~ z)
by FINSEQ_4:67;
then A40:
(g /. (len g)) `1 = E-bound (L~ z)
by EUCLID:52;
(
M is_in_the_area_of z &
<*(NE-corner (L~ z))*> is_in_the_area_of z )
by A19, A26, Th21, Th22, Th25;
then A41:
g is_in_the_area_of z
by Th24;
(LSeg ((M /. (len M)),(NE-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. (len M)),(NE-corner (L~ z)))) /\ (L~ z)
by A11, A8, A14, JORDAN4:35, XBOOLE_1:26;
then A42:
(LSeg ((M /. (len M)),(NE-corner (L~ z)))) /\ (L~ h) c= {(M /. (len M))}
by A27, PSCOMP_1:43;
g /. 1 =
M /. 1
by A25, FINSEQ_4:68
.=
z /. 1
by A19, A26, A31, Th8
;
then
(g /. 1) `1 = W-bound (L~ z)
by A1, A37, EUCLID:52;
then
g is_a_h.c._for z
by A41, A40;
hence
contradiction
by A17, A28, A30, A39, A42, Th29, ZFMISC_1:125;
verum end; suppose that A43:
NW-corner (L~ z) <> N-min (L~ z)
and A44:
NE-corner (L~ z) = N-max (L~ z)
;
contradictionreconsider g =
<*(NW-corner (L~ z))*> ^ M as
S-Sequence_in_R2 by A1, A19, A26, A31, A43, Th66;
(
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 /. ((N-max (L~ z)) .. z)
by A19, A26, Th9
.=
N-max (L~ z)
by A18, FINSEQ_5:38
;
then A45:
(g /. (len g)) `1 = E-bound (L~ z)
by A44, EUCLID:52;
A46:
(
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 A47:
(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 A11, A8, A14, JORDAN4:35, XBOOLE_1:26;
then A48:
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)}
by A1, A32, PSCOMP_1:43;
A49:
M /. 1
in L~ M
by A29, A23, JORDAN3:1;
(
M is_in_the_area_of z &
<*(NW-corner (L~ z))*> is_in_the_area_of z )
by A19, A26, Th21, Th22, Th26;
then
g is_in_the_area_of z
by Th24;
then
g is_a_h.c._for z
by A47, A45;
hence
contradiction
by A17, A28, A46, A48, A49, Th29, ZFMISC_1:125;
verum end; suppose A50:
(
NW-corner (L~ z) <> N-min (L~ z) &
NE-corner (L~ z) <> N-max (L~ z) )
;
contradictionset K =
<*(NW-corner (L~ z))*> ^ M;
reconsider g =
(<*(NW-corner (L~ z))*> ^ M) ^ <*(NE-corner (L~ z))*> as
S-Sequence_in_R2 by A1, A19, A20, A26, A31, A50, Lm1;
1
in dom (<*(NW-corner (L~ z))*> ^ M)
by FINSEQ_5:6;
then g /. 1 =
(<*(NW-corner (L~ z))*> ^ M) /. 1
by FINSEQ_4:68
.=
NW-corner (L~ z)
by FINSEQ_5:15
;
then A51:
(g /. 1) `1 = W-bound (L~ z)
by EUCLID:52;
len g =
(len (<*(NW-corner (L~ z))*> ^ M)) + (len <*(NE-corner (L~ z))*>)
by FINSEQ_1:22
.=
(len (<*(NW-corner (L~ z))*> ^ M)) + 1
by FINSEQ_1:39
;
then
g /. (len g) = NE-corner (L~ z)
by FINSEQ_4:67;
then A52:
(g /. (len g)) `1 = E-bound (L~ z)
by EUCLID:52;
(
M is_in_the_area_of z &
<*(NW-corner (L~ z))*> is_in_the_area_of z )
by A19, A26, Th21, Th22, Th26;
then A53:
<*(NW-corner (L~ z))*> ^ M is_in_the_area_of z
by Th24;
<*(NE-corner (L~ z))*> is_in_the_area_of z
by Th25;
then
g is_in_the_area_of z
by A53, Th24;
then A54:
g is_a_h.c._for z
by A51, A52;
len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>)
by FINSEQ_1:22;
then
len (<*(NW-corner (L~ z))*> ^ M) >= len M
by NAT_1:11;
then
len (<*(NW-corner (L~ z))*> ^ M) >= 2
by A29, A23, XXREAL_0:2;
then A55:
(<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) in L~ (<*(NW-corner (L~ z))*> ^ M)
by JORDAN3:1;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z)
by A11, A8, A14, JORDAN4:35, XBOOLE_1:26;
then A56:
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)}
by A1, A32, PSCOMP_1:43;
(
L~ (<*(NW-corner (L~ z))*> ^ M) = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) &
M /. 1
in L~ M )
by A29, A23, JORDAN3:1, SPPOL_2:20;
then A57:
L~ (<*(NW-corner (L~ z))*> ^ M) misses L~ h
by A28, A56, ZFMISC_1:125;
(
len M in dom M &
len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>) )
by FINSEQ_1:22, FINSEQ_5:6;
then A58:
(<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) =
M /. (len M)
by FINSEQ_4:69
.=
z /. ((N-max (L~ z)) .. z)
by A19, A26, Th9
.=
N-max (L~ z)
by A18, FINSEQ_5:38
;
(LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) /\ (L~ h) c= (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) /\ (L~ z)
by A11, A8, A14, JORDAN4:35, XBOOLE_1:26;
then A59:
(LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) /\ (L~ h) c= {((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)))}
by A58, PSCOMP_1:43;
(
len g >= 2 &
L~ g = (L~ (<*(NW-corner (L~ z))*> ^ M)) \/ (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) )
by SPPOL_2:19, TOPREAL1:def 8;
hence
contradiction
by A17, A54, A57, A55, A59, Th29, ZFMISC_1:125;
verum end;