let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( 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)
and
A3:
(E-min (L~ z)) .. z >= (S-max (L~ z)) .. z
; :: thesis: contradiction
A4:
E-min (L~ z) in rng z
by Th49;
A5:
S-max (L~ z) in rng z
by Th46;
A6:
(E-min (L~ z)) .. z in dom z
by A4, FINSEQ_4:30;
then A7:
( 1 <= (E-min (L~ z)) .. z & (E-min (L~ z)) .. z <= len z )
by FINSEQ_3:27;
A8:
(S-max (L~ z)) .. z in dom z
by A5, FINSEQ_4:30;
then A9:
( 1 <= (S-max (L~ z)) .. z & (S-max (L~ z)) .. z <= len z )
by FINSEQ_3:27;
A10: z /. ((S-max (L~ z)) .. z) =
z . ((S-max (L~ z)) .. z)
by A8, PARTFUN1:def 8
.=
S-max (L~ z)
by A5, FINSEQ_4:29
;
A11:
z /. (len z) = N-min (L~ z)
by A1, FINSEQ_6:def 1;
A12: z /. ((E-min (L~ z)) .. z) =
z . ((E-min (L~ z)) .. z)
by A6, PARTFUN1:def 8
.=
E-min (L~ z)
by A4, FINSEQ_4:29
;
then A13:
(E-min (L~ z)) .. z > (S-max (L~ z)) .. z
by A2, A3, A10, XXREAL_0:1;
then A14:
(E-min (L~ z)) .. z > 1
by A9, XXREAL_0:2;
A15:
len z in dom z
by FINSEQ_5:6;
A16:
z /. 1 = z /. (len z)
by FINSEQ_6:def 1;
A17:
2 <= len z
by REALSET1:13;
then A18:
2 in dom z
by FINSEQ_3:27;
z /. 2 in N-most (L~ z)
by A1, Th34;
then A19: (z /. 2) `2 =
(N-min (L~ z)) `2
by PSCOMP_1:98
.=
N-bound (L~ z)
by EUCLID:56
;
A20:
(z /. 1) `2 = N-bound (L~ z)
by A1, EUCLID:56;
A21:
(S-max (L~ z)) .. z <> 0
by A8, FINSEQ_3:27;
A22:
(z /. ((S-max (L~ z)) .. z)) `2 = S-bound (L~ z)
by A10, EUCLID:56;
S-bound (L~ z) < N-bound (L~ z)
by TOPREAL5:22;
then A23:
(S-max (L~ z)) .. z > 2
by A19, A20, A21, A22, NAT_1:27;
then reconsider h = mid z,((S-max (L~ z)) .. z),2 as S-Sequence_in_R2 by A9, Th41;
A24:
len h >= 2
by TOPREAL1:def 10;
A25:
h is_in_the_area_of z
by A8, A18, Th25, Th26;
h /. 1 = S-max (L~ z)
by A8, A10, A18, Th12;
then A26:
(h /. 1) `2 = S-bound (L~ z)
by EUCLID:56;
h /. (len h) = z /. 2
by A8, A18, Th13;
then A27:
h is_a_v.c._for z
by A19, A25, A26, Def3;
N-max (L~ z) in L~ z
by SPRECT_1:13;
then
(N-max (L~ z)) `1 <= E-bound (L~ z)
by PSCOMP_1:71;
then
(N-min (L~ z)) `1 < E-bound (L~ z)
by Th55, XXREAL_0:2;
then
(N-min (L~ z)) `1 < (E-min (L~ z)) `1
by EUCLID:56;
then A28:
(E-min (L~ z)) .. z < len z
by A7, A11, A12, XXREAL_0:1;
then reconsider M = mid z,(len z),((E-min (L~ z)) .. z) as S-Sequence_in_R2 by A14, Th41;
A29:
L~ M misses L~ h
by A7, A13, A23, Th53;
A30: M /. (len M) =
z /. ((E-min (L~ z)) .. z)
by A6, A15, Th13
.=
E-min (L~ z)
by A4, FINSEQ_5:41
;
A31:
len M = ((len z) -' ((E-min (L~ z)) .. z)) + 1
by A7, JORDAN4:21;
((E-min (L~ z)) .. z) + 1 <= len z
by A28, NAT_1:13;
then
(len z) - ((E-min (L~ z)) .. z) >= 1
by XREAL_1:21;
then
(len z) -' ((E-min (L~ z)) .. z) >= 1
by NAT_D:39;
then A32:
((len z) -' ((E-min (L~ z)) .. z)) + 1 >= 1 + 1
by XREAL_1:8;
A33:
M /. 1 = z /. 1
by A6, A15, A16, Th12;
per cases
( NW-corner (L~ z) = N-min (L~ z) or NW-corner (L~ z) <> N-min (L~ z) )
;
suppose A34:
NW-corner (L~ z) = N-min (L~ z)
;
:: thesis: contradictionA35:
M is_in_the_area_of z
by A6, A15, Th25, Th26;
M /. 1
= z /. (len z)
by A6, A15, Th12;
then A36:
(M /. 1) `1 = W-bound (L~ z)
by A1, A16, A34, EUCLID:56;
(M /. (len M)) `1 = E-bound (L~ z)
by A30, EUCLID:56;
then
M is_a_h.c._for z
by A35, A36, Def2;
hence
contradiction
by A24, A27, A29, A31, A32, Th33;
:: thesis: verum end; suppose
NW-corner (L~ z) <> N-min (L~ z)
;
:: thesis: contradictionthen reconsider g =
<*(NW-corner (L~ z))*> ^ M as
S-Sequence_in_R2 by A1, A6, A15, A16, Th70;
A37:
len g >= 2
by TOPREAL1:def 10;
A38:
M is_in_the_area_of z
by A6, A15, Th25, Th26;
<*(NW-corner (L~ z))*> is_in_the_area_of z
by Th30;
then A39:
g is_in_the_area_of z
by A38, Th28;
g /. 1
= NW-corner (L~ z)
by FINSEQ_5:16;
then A40:
(g /. 1) `1 = W-bound (L~ z)
by EUCLID:56;
A41:
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 A41, FINSEQ_4:84
.=
z /. ((E-min (L~ z)) .. z)
by A6, A15, Th13
.=
E-min (L~ z)
by A4, FINSEQ_5:41
;
then
(g /. (len g)) `1 = E-bound (L~ z)
by EUCLID:56;
then A42:
g is_a_h.c._for z
by A39, A40, Def2;
A43:
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 A9, A17, JORDAN4:47, XBOOLE_1:26;
then A44:
(LSeg (M /. 1),(NW-corner (L~ z))) /\ (L~ h) c= {(M /. 1)}
by A1, A33, PSCOMP_1:102;
M /. 1
in L~ M
by A31, A32, JORDAN3:34;
hence
contradiction
by A24, A27, A29, A37, A42, A43, A44, Th33, ZFMISC_1:149;
:: thesis: verum end;