let z be constant standard clockwise_oriented special_circular_sequence; ( z /. 1 = N-min (L~ z) implies (S-max (L~ z)) .. z < (S-min (L~ z)) .. z )
set i1 = (S-max (L~ z)) .. z;
set i2 = (S-min (L~ z)) .. z;
set j = (N-max (L~ z)) .. z;
assume that
A1:
z /. 1 = N-min (L~ z)
and
A2:
(S-max (L~ z)) .. z >= (S-min (L~ z)) .. z
; contradiction
A3:
z /. 1 = z /. (len z)
by FINSEQ_6:def 1;
A4:
S-min (L~ z) in rng z
by Th41;
then A5:
(S-min (L~ z)) .. z in dom z
by FINSEQ_4:20;
then A6:
(S-min (L~ z)) .. z <= len z
by FINSEQ_3:25;
A7:
1 <= (S-min (L~ z)) .. z
by A5, FINSEQ_3:25;
A8:
S-max (L~ z) in rng z
by Th42;
then A9:
(S-max (L~ z)) .. z in dom z
by FINSEQ_4:20;
then A10: z /. ((S-max (L~ z)) .. z) =
z . ((S-max (L~ z)) .. z)
by PARTFUN1:def 6
.=
S-max (L~ z)
by A8, FINSEQ_4:19
;
A11:
(S-max (L~ z)) .. z <= len z
by A9, FINSEQ_3:25;
( (N-min (L~ z)) `2 = N-bound (L~ z) & (S-max (L~ z)) `2 = S-bound (L~ z) )
by EUCLID:52;
then
N-min (L~ z) <> S-max (L~ z)
by TOPREAL5:16;
then A12:
(S-max (L~ z)) .. z < len z
by A1, A11, A10, A3, XXREAL_0:1;
then
((S-max (L~ z)) .. z) + 1 <= len z
by NAT_1:13;
then
(len z) - ((S-max (L~ z)) .. z) >= 1
by XREAL_1:19;
then
(len z) -' ((S-max (L~ z)) .. z) >= 1
by NAT_D:39;
then A13:
((len z) -' ((S-max (L~ z)) .. z)) + 1 >= 1 + 1
by XREAL_1:6;
A14:
N-max (L~ z) in rng z
by Th40;
then A15:
(N-max (L~ z)) .. z in dom z
by FINSEQ_4:20;
then A16:
1 <= (N-max (L~ z)) .. z
by FINSEQ_3:25;
then
(S-max (L~ z)) .. z > 1
by A1, Lm5, XXREAL_0:2;
then reconsider M = mid (z,(len z),((S-max (L~ z)) .. z)) as S-Sequence_in_R2 by A12, Th37;
A17: z /. ((N-max (L~ z)) .. z) =
z . ((N-max (L~ z)) .. z)
by A15, PARTFUN1:def 6
.=
N-max (L~ z)
by A14, FINSEQ_4:19
;
then A18:
(z /. ((N-max (L~ z)) .. z)) `2 = N-bound (L~ z)
by EUCLID:52;
N-min (L~ z) <> N-max (L~ z)
by Th52;
then A19:
1 < (N-max (L~ z)) .. z
by A1, A16, A17, XXREAL_0:1;
A20:
len z in dom z
by FINSEQ_5:6;
then A21:
M /. 1 = z /. (len z)
by A9, Th8;
1 <= (S-max (L~ z)) .. z
by A9, FINSEQ_3:25;
then A22:
len M = ((len z) -' ((S-max (L~ z)) .. z)) + 1
by A11, FINSEQ_6:187;
then A23:
M /. (len M) in L~ M
by A13, JORDAN3:1;
A24:
1 in dom M
by FINSEQ_5:6;
A25:
(N-max (L~ z)) .. z <= len z
by A15, FINSEQ_3:25;
A26:
(S-min (L~ z)) .. z > (N-max (L~ z)) .. z
by A1, Lm6;
then reconsider h = mid (z,((S-min (L~ z)) .. z),((N-max (L~ z)) .. z)) as S-Sequence_in_R2 by A6, A19, Th37;
A27: z /. ((S-min (L~ z)) .. z) =
z . ((S-min (L~ z)) .. z)
by A5, PARTFUN1:def 6
.=
S-min (L~ z)
by A4, FINSEQ_4:19
;
then
h /. 1 = S-min (L~ z)
by A5, A15, Th8;
then A28:
(h /. 1) `2 = S-bound (L~ z)
by EUCLID:52;
( h is_in_the_area_of z & h /. (len h) = z /. ((N-max (L~ z)) .. z) )
by A5, A15, Th9, Th21, Th22;
then A29:
( len h >= 2 & h is_a_v.c._for z )
by A18, A28, TOPREAL1:def 8;
S-min (L~ z) <> S-max (L~ z)
by Th56;
then
(S-max (L~ z)) .. z > (S-min (L~ z)) .. z
by A2, A27, A10, XXREAL_0:1;
then A30:
L~ h misses L~ M
by A11, A26, A19, Th49;
A31:
M /. (len M) = S-max (L~ z)
by A9, A10, A20, Th9;
per cases
( ( NW-corner (L~ z) = N-min (L~ z) & SE-corner (L~ z) = S-max (L~ z) ) or ( NW-corner (L~ z) = N-min (L~ z) & SE-corner (L~ z) <> S-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & SE-corner (L~ z) = S-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & SE-corner (L~ z) <> S-max (L~ z) ) )
;
suppose A32:
(
NW-corner (L~ z) = N-min (L~ z) &
SE-corner (L~ z) = S-max (L~ z) )
;
contradictionA33:
M is_in_the_area_of z
by A9, A20, Th21, Th22;
(
(M /. 1) `1 = W-bound (L~ z) &
(M /. (len M)) `1 = E-bound (L~ z) )
by A1, A3, A31, A21, A32, EUCLID:52;
then
M is_a_h.c._for z
by A33;
hence
contradiction
by A29, A30, A22, A13, Th29;
verum end; suppose that A34:
NW-corner (L~ z) = N-min (L~ z)
and A35:
SE-corner (L~ z) <> S-max (L~ z)
;
contradictionreconsider g =
M ^ <*(SE-corner (L~ z))*> as
S-Sequence_in_R2 by A9, A10, A20, A35, Th64;
A36:
(
len g >= 2 &
L~ g = (L~ M) \/ (LSeg ((M /. (len M)),(SE-corner (L~ z)))) )
by SPPOL_2:19, TOPREAL1:def 8;
len g =
(len M) + (len <*(SE-corner (L~ z))*>)
by FINSEQ_1:22
.=
(len M) + 1
by FINSEQ_1:39
;
then
g /. (len g) = SE-corner (L~ z)
by FINSEQ_4:67;
then A37:
(g /. (len g)) `1 = E-bound (L~ z)
by EUCLID:52;
(
M is_in_the_area_of z &
<*(SE-corner (L~ z))*> is_in_the_area_of z )
by A9, A20, Th21, Th22, Th27;
then A38:
g is_in_the_area_of z
by Th24;
(LSeg ((M /. (len M)),(SE-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. (len M)),(SE-corner (L~ z)))) /\ (L~ z)
by A7, A6, A16, A25, JORDAN4:35, XBOOLE_1:26;
then A39:
(LSeg ((M /. (len M)),(SE-corner (L~ z)))) /\ (L~ h) c= {(M /. (len M))}
by A31, PSCOMP_1:59;
g /. 1 =
M /. 1
by A24, FINSEQ_4:68
.=
z /. 1
by A9, A3, A20, Th8
;
then
(g /. 1) `1 = W-bound (L~ z)
by A1, A34, EUCLID:52;
then
g is_a_h.c._for z
by A38, A37;
hence
contradiction
by A29, A30, A23, A36, A39, Th29, ZFMISC_1:125;
verum end; suppose that A40:
NW-corner (L~ z) <> N-min (L~ z)
and A41:
SE-corner (L~ z) = S-max (L~ z)
;
contradictionreconsider g =
<*(NW-corner (L~ z))*> ^ M as
S-Sequence_in_R2 by A1, A9, A3, A20, A40, 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
.=
S-max (L~ z)
by A9, A10, A20, Th9
;
then A42:
(g /. (len g)) `1 = E-bound (L~ z)
by A41, EUCLID:52;
A43:
(
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 A44:
(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 A7, A6, A16, A25, JORDAN4:35, XBOOLE_1:26;
then A45:
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)}
by A1, A3, A21, PSCOMP_1:43;
A46:
M /. 1
in L~ M
by A22, A13, JORDAN3:1;
(
M is_in_the_area_of z &
<*(NW-corner (L~ z))*> is_in_the_area_of z )
by A9, A20, Th21, Th22, Th26;
then
g is_in_the_area_of z
by Th24;
then
g is_a_h.c._for z
by A44, A42;
hence
contradiction
by A29, A30, A43, A45, A46, Th29, ZFMISC_1:125;
verum end; suppose A47:
(
NW-corner (L~ z) <> N-min (L~ z) &
SE-corner (L~ z) <> S-max (L~ z) )
;
contradictionset K =
<*(NW-corner (L~ z))*> ^ M;
reconsider g =
(<*(NW-corner (L~ z))*> ^ M) ^ <*(SE-corner (L~ z))*> as
S-Sequence_in_R2 by A1, A9, A10, A3, A20, A47, Lm3;
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 A48:
(g /. 1) `1 = W-bound (L~ z)
by EUCLID:52;
len g =
(len (<*(NW-corner (L~ z))*> ^ M)) + (len <*(SE-corner (L~ z))*>)
by FINSEQ_1:22
.=
(len (<*(NW-corner (L~ z))*> ^ M)) + 1
by FINSEQ_1:39
;
then
g /. (len g) = SE-corner (L~ z)
by FINSEQ_4:67;
then A49:
(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 A9, A20, Th21, Th22, Th26;
then A50:
<*(NW-corner (L~ z))*> ^ M is_in_the_area_of z
by Th24;
<*(SE-corner (L~ z))*> is_in_the_area_of z
by Th27;
then
g is_in_the_area_of z
by A50, Th24;
then A51:
g is_a_h.c._for z
by A48, A49;
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 A22, A13, XXREAL_0:2;
then A52:
(<*(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 A7, A6, A16, A25, JORDAN4:35, XBOOLE_1:26;
then A53:
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)}
by A1, A3, A21, PSCOMP_1:43;
(
L~ (<*(NW-corner (L~ z))*> ^ M) = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) &
M /. 1
in L~ M )
by A22, A13, JORDAN3:1, SPPOL_2:20;
then A54:
L~ (<*(NW-corner (L~ z))*> ^ M) misses L~ h
by A30, A53, 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 A55:
(<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) =
M /. (len M)
by FINSEQ_4:69
.=
z /. ((S-max (L~ z)) .. z)
by A9, A20, Th9
.=
S-max (L~ z)
by A8, FINSEQ_5:38
;
(LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(SE-corner (L~ z)))) /\ (L~ h) c= (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(SE-corner (L~ z)))) /\ (L~ z)
by A7, A6, A16, A25, JORDAN4:35, XBOOLE_1:26;
then A56:
(LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(SE-corner (L~ z)))) /\ (L~ h) c= {((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)))}
by A55, PSCOMP_1:59;
(
len g >= 2 &
L~ g = (L~ (<*(NW-corner (L~ z))*> ^ M)) \/ (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(SE-corner (L~ z)))) )
by SPPOL_2:19, TOPREAL1:def 8;
hence
contradiction
by A29, A51, A54, A52, A56, Th29, ZFMISC_1:125;
verum end;