let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( z /. 1 = N-min (L~ z) implies (N-max (L~ z)) .. z < (S-max (L~ z)) .. z )
set i1 = (N-max (L~ z)) .. z;
set i2 = (S-max (L~ z)) .. z;
assume that
A1:
z /. 1 = N-min (L~ z)
and
A2:
(N-max (L~ z)) .. z >= (S-max (L~ z)) .. z
; :: thesis: contradiction
A3:
N-max (L~ z) in rng z
by Th44;
A4:
S-max (L~ z) in rng z
by Th46;
A5:
(N-max (L~ z)) .. z in dom z
by A3, FINSEQ_4:30;
then A6:
( 1 <= (N-max (L~ z)) .. z & (N-max (L~ z)) .. z <= len z )
by FINSEQ_3:27;
A7:
(S-max (L~ z)) .. z in dom z
by A4, FINSEQ_4:30;
then A8:
( 1 <= (S-max (L~ z)) .. z & (S-max (L~ z)) .. z <= len z )
by FINSEQ_3:27;
A9: z /. ((S-max (L~ z)) .. z) =
z . ((S-max (L~ z)) .. z)
by A7, PARTFUN1:def 8
.=
S-max (L~ z)
by A4, FINSEQ_4:29
;
A10:
z /. (len z) = N-min (L~ z)
by A1, FINSEQ_6:def 1;
A11: z /. ((N-max (L~ z)) .. z) =
z . ((N-max (L~ z)) .. z)
by A5, PARTFUN1:def 8
.=
N-max (L~ z)
by A3, FINSEQ_4:29
;
( (N-max (L~ z)) `2 = N-bound (L~ z) & (S-max (L~ z)) `2 = S-bound (L~ z) )
by EUCLID:56;
then
z /. ((N-max (L~ z)) .. z) <> z /. ((S-max (L~ z)) .. z)
by A9, A11, TOPREAL5:22;
then A12:
(N-max (L~ z)) .. z > (S-max (L~ z)) .. z
by A2, XXREAL_0:1;
then A13:
(N-max (L~ z)) .. z > 1
by A8, XXREAL_0:2;
A14:
len z in dom z
by FINSEQ_5:6;
A15:
z /. 1 = z /. (len z)
by FINSEQ_6:def 1;
A16:
2 <= len z
by REALSET1:13;
then A17:
2 in dom z
by FINSEQ_3:27;
z /. 2 in N-most (L~ z)
by A1, Th34;
then A18: (z /. 2) `2 =
(N-min (L~ z)) `2
by PSCOMP_1:98
.=
N-bound (L~ z)
by EUCLID:56
;
A19:
(z /. 1) `2 = N-bound (L~ z)
by A1, EUCLID:56;
A20:
(S-max (L~ z)) .. z <> 0
by A7, FINSEQ_3:27;
(z /. ((S-max (L~ z)) .. z)) `2 = S-bound (L~ z)
by A9, EUCLID:56;
then
( (S-max (L~ z)) .. z <> 1 & (S-max (L~ z)) .. z <> 2 )
by A18, A19, TOPREAL5:22;
then A21:
(S-max (L~ z)) .. z > 2
by A20, NAT_1:27;
then reconsider h = mid z,((S-max (L~ z)) .. z),2 as S-Sequence_in_R2 by A8, Th41;
A22:
h is_in_the_area_of z
by A7, A17, Th25, Th26;
h /. 1 = S-max (L~ z)
by A7, A9, A17, Th12;
then A23:
(h /. 1) `2 = S-bound (L~ z)
by EUCLID:56;
(h /. (len h)) `2 = N-bound (L~ z)
by A7, A17, A18, Th13;
then A24:
h is_a_v.c._for z
by A22, A23, Def3;
N-min (L~ z) <> N-max (L~ z)
by Th56;
then A25:
(N-max (L~ z)) .. z < len z
by A6, A10, A11, XXREAL_0:1;
then reconsider M = mid z,(len z),((N-max (L~ z)) .. z) as S-Sequence_in_R2 by A13, Th41;
A26:
L~ M misses L~ h
by A6, A12, A21, Th53;
A27: M /. (len M) =
z /. ((N-max (L~ z)) .. z)
by A5, A14, Th13
.=
N-max (L~ z)
by A3, FINSEQ_5:41
;
A28:
len M = ((len z) -' ((N-max (L~ z)) .. z)) + 1
by A6, JORDAN4:21;
((N-max (L~ z)) .. z) + 1 <= len z
by A25, NAT_1:13;
then
(len z) - ((N-max (L~ z)) .. z) >= 1
by XREAL_1:21;
then
(len z) -' ((N-max (L~ z)) .. z) >= 1
by NAT_D:39;
then A29:
((len z) -' ((N-max (L~ z)) .. z)) + 1 >= 1 + 1
by XREAL_1:8;
then A30:
M /. (len M) in L~ M
by A28, JORDAN3:34;
A31:
2 <= len h
by TOPREAL1:def 10;
A32:
1 in dom M
by FINSEQ_5:6;
A33:
M /. 1 = z /. 1
by A5, A14, A15, Th12;
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 A34:
NW-corner (L~ z) = N-min (L~ z)
and A35:
NE-corner (L~ z) = N-max (L~ z)
;
:: thesis: contradictionA36:
M is_in_the_area_of z
by A5, A14, Th25, Th26;
M /. 1
= z /. (len z)
by A5, A14, Th12;
then A37:
(M /. 1) `1 = W-bound (L~ z)
by A1, A15, A34, EUCLID:56;
(M /. (len M)) `1 = E-bound (L~ z)
by A27, A35, EUCLID:56;
then
M is_a_h.c._for z
by A36, A37, Def2;
hence
contradiction
by A24, A26, A28, A29, A31, Th33;
:: thesis: verum end; suppose that A38:
NW-corner (L~ z) = N-min (L~ z)
and A39:
NE-corner (L~ z) <> N-max (L~ z)
;
:: thesis: contradictionreconsider g =
M ^ <*(NE-corner (L~ z))*> as
S-Sequence_in_R2 by A5, A11, A14, A39, Th66;
A40:
len g >= 2
by TOPREAL1:def 10;
A41:
M is_in_the_area_of z
by A5, A14, Th25, Th26;
<*(NE-corner (L~ z))*> is_in_the_area_of z
by Th29;
then A42:
g is_in_the_area_of z
by A41, Th28;
g /. 1 =
M /. 1
by A32, FINSEQ_4:83
.=
z /. 1
by A5, A14, A15, Th12
;
then A43:
(g /. 1) `1 = W-bound (L~ z)
by A1, A38, EUCLID:56;
len g =
(len M) + (len <*(NE-corner (L~ z))*>)
by FINSEQ_1:35
.=
(len M) + 1
by FINSEQ_1:56
;
then
g /. (len g) = NE-corner (L~ z)
by FINSEQ_4:82;
then
(g /. (len g)) `1 = E-bound (L~ z)
by EUCLID:56;
then A44:
g is_a_h.c._for z
by A42, A43, Def2;
A45:
L~ g = (L~ M) \/ (LSeg (M /. (len M)),(NE-corner (L~ z)))
by SPPOL_2:19;
(LSeg (M /. (len M)),(NE-corner (L~ z))) /\ (L~ h) c= (LSeg (M /. (len M)),(NE-corner (L~ z))) /\ (L~ z)
by A8, A16, JORDAN4:47, XBOOLE_1:26;
then
(LSeg (M /. (len M)),(NE-corner (L~ z))) /\ (L~ h) c= {(M /. (len M))}
by A27, PSCOMP_1:102;
hence
contradiction
by A24, A26, A30, A31, A40, A44, A45, Th33, ZFMISC_1:149;
:: thesis: verum end; suppose that A46:
NW-corner (L~ z) <> N-min (L~ z)
and A47:
NE-corner (L~ z) = N-max (L~ z)
;
:: thesis: contradictionreconsider g =
<*(NW-corner (L~ z))*> ^ M as
S-Sequence_in_R2 by A1, A5, A14, A15, A46, Th70;
A48:
len g >= 2
by TOPREAL1:def 10;
A49:
M is_in_the_area_of z
by A5, A14, Th25, Th26;
<*(NW-corner (L~ z))*> is_in_the_area_of z
by Th30;
then A50:
g is_in_the_area_of z
by A49, Th28;
g /. 1
= NW-corner (L~ z)
by FINSEQ_5:16;
then A51:
(g /. 1) `1 = W-bound (L~ z)
by EUCLID:56;
A52:
len M in dom M
by FINSEQ_5:6;
len g = (len M) + (len <*(NW-corner (L~ z))*>)
by FINSEQ_1:35;
then g /. (len g) =
M /. (len M)
by A52, FINSEQ_4:84
.=
z /. ((N-max (L~ z)) .. z)
by A5, A14, Th13
.=
N-max (L~ z)
by A3, FINSEQ_5:41
;
then
(g /. (len g)) `1 = E-bound (L~ z)
by A47, EUCLID:56;
then A53:
g is_a_h.c._for z
by A50, A51, Def2;
A54:
L~ g = (L~ M) \/ (LSeg (NW-corner (L~ z)),(M /. 1))
by SPPOL_2:20;
(LSeg (M /. 1),(NW-corner (L~ z))) /\ (L~ h) c= (LSeg (M /. 1),(NW-corner (L~ z))) /\ (L~ z)
by A8, A16, JORDAN4:47, XBOOLE_1:26;
then A55:
(LSeg (M /. 1),(NW-corner (L~ z))) /\ (L~ h) c= {(M /. 1)}
by A1, A33, PSCOMP_1:102;
M /. 1
in L~ M
by A28, A29, JORDAN3:34;
hence
contradiction
by A24, A26, A31, A48, A53, A54, A55, Th33, ZFMISC_1:149;
:: thesis: verum end; suppose that A56:
NW-corner (L~ z) <> N-min (L~ z)
and A57:
NE-corner (L~ z) <> N-max (L~ z)
;
:: thesis: 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, A5, A11, A14, A15, A56, A57, Lm1;
A58:
len g >= 2
by TOPREAL1:def 10;
A59:
M is_in_the_area_of z
by A5, A14, Th25, Th26;
<*(NW-corner (L~ z))*> is_in_the_area_of z
by Th30;
then A60:
<*(NW-corner (L~ z))*> ^ M is_in_the_area_of z
by A59, Th28;
<*(NE-corner (L~ z))*> is_in_the_area_of z
by Th29;
then A61:
g is_in_the_area_of z
by A60, Th28;
1
in dom (<*(NW-corner (L~ z))*> ^ M)
by FINSEQ_5:6;
then g /. 1 =
(<*(NW-corner (L~ z))*> ^ M) /. 1
by FINSEQ_4:83
.=
NW-corner (L~ z)
by FINSEQ_5:16
;
then A62:
(g /. 1) `1 = W-bound (L~ z)
by EUCLID:56;
len g =
(len (<*(NW-corner (L~ z))*> ^ M)) + (len <*(NE-corner (L~ z))*>)
by FINSEQ_1:35
.=
(len (<*(NW-corner (L~ z))*> ^ M)) + 1
by FINSEQ_1:56
;
then
g /. (len g) = NE-corner (L~ z)
by FINSEQ_4:82;
then
(g /. (len g)) `1 = E-bound (L~ z)
by EUCLID:56;
then A63:
g is_a_h.c._for z
by A61, A62, Def2;
A64:
L~ (<*(NW-corner (L~ z))*> ^ M) = (L~ M) \/ (LSeg (NW-corner (L~ z)),(M /. 1))
by SPPOL_2:20;
(LSeg (M /. 1),(NW-corner (L~ z))) /\ (L~ h) c= (LSeg (M /. 1),(NW-corner (L~ z))) /\ (L~ z)
by A8, A16, JORDAN4:47, XBOOLE_1:26;
then A65:
(LSeg (M /. 1),(NW-corner (L~ z))) /\ (L~ h) c= {(M /. 1)}
by A1, A33, PSCOMP_1:102;
M /. 1
in L~ M
by A28, A29, JORDAN3:34;
then A66:
L~ (<*(NW-corner (L~ z))*> ^ M) misses L~ h
by A26, A64, A65, ZFMISC_1:149;
len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>)
by FINSEQ_1:35;
then
len (<*(NW-corner (L~ z))*> ^ M) >= len M
by NAT_1:11;
then
len (<*(NW-corner (L~ z))*> ^ M) >= 2
by A28, A29, XXREAL_0:2;
then A67:
(<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) in L~ (<*(NW-corner (L~ z))*> ^ M)
by JORDAN3:34;
A68:
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;
A69:
len M in dom M
by FINSEQ_5:6;
len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>)
by FINSEQ_1:35;
then A70:
(<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) =
M /. (len M)
by A69, FINSEQ_4:84
.=
z /. ((N-max (L~ z)) .. z)
by A5, A14, Th13
.=
N-max (L~ z)
by A3, FINSEQ_5:41
;
(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 A8, A16, JORDAN4:47, XBOOLE_1:26;
then
(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 A70, PSCOMP_1:102;
hence
contradiction
by A24, A31, A58, A63, A66, A67, A68, Th33, ZFMISC_1:149;
:: thesis: verum end;