let z be non constant standard clockwise_oriented special_circular_sequence; ( z /. 1 = N-min (L~ z) & N-min (L~ z) <> W-max (L~ z) implies (W-min (L~ z)) .. z < (W-max (L~ z)) .. z )
set i1 = (W-min (L~ z)) .. z;
set i2 = (W-max (L~ z)) .. z;
set j = (E-min (L~ z)) .. z;
assume that
A1:
z /. 1 = N-min (L~ z)
and
A2:
N-min (L~ z) <> W-max (L~ z)
and
A3:
(W-min (L~ z)) .. z >= (W-max (L~ z)) .. z
; contradiction
A4:
(W-max (L~ z)) .. z > (E-min (L~ z)) .. z
by A1, A2, Lm10;
A5:
E-min (L~ z) in rng z
by Th49;
then A6:
(E-min (L~ z)) .. z in dom z
by FINSEQ_4:30;
then A7: z /. ((E-min (L~ z)) .. z) =
z . ((E-min (L~ z)) .. z)
by PARTFUN1:def 8
.=
E-min (L~ z)
by A5, FINSEQ_4:29
;
then A8:
(z /. ((E-min (L~ z)) .. z)) `1 = E-bound (L~ z)
by EUCLID:56;
A9:
(E-min (L~ z)) .. z <= len z
by A6, FINSEQ_3:27;
A10:
z /. (len z) = N-min (L~ z)
by A1, FINSEQ_6:def 1;
A11:
W-max (L~ z) in rng z
by Th48;
then A12:
(W-max (L~ z)) .. z in dom z
by FINSEQ_4:30;
then A13:
1 <= (W-max (L~ z)) .. z
by FINSEQ_3:27;
A14:
W-min (L~ z) in rng z
by Th47;
then A15:
(W-min (L~ z)) .. z in dom z
by FINSEQ_4:30;
then A16: z /. ((W-min (L~ z)) .. z) =
z . ((W-min (L~ z)) .. z)
by PARTFUN1:def 8
.=
W-min (L~ z)
by A14, FINSEQ_4:29
;
A17:
(W-min (L~ z)) .. z <= len z
by A15, FINSEQ_3:27;
( W-max (L~ z) in L~ z & (N-min (L~ z)) `2 = N-bound (L~ z) )
by EUCLID:56, SPRECT_1:15;
then
(W-max (L~ z)) `2 <= (N-min (L~ z)) `2
by PSCOMP_1:71;
then
( z /. 1 = z /. (len z) & N-min (L~ z) <> W-min (L~ z) )
by Th61, FINSEQ_6:def 1;
then A18:
(W-min (L~ z)) .. z < len z
by A1, A17, A16, XXREAL_0:1;
then
((W-min (L~ z)) .. z) + 1 <= len z
by NAT_1:13;
then
(len z) - ((W-min (L~ z)) .. z) >= 1
by XREAL_1:21;
then
(len z) -' ((W-min (L~ z)) .. z) >= 1
by NAT_D:39;
then A19:
((len z) -' ((W-min (L~ z)) .. z)) + 1 >= 1 + 1
by XREAL_1:8;
A20:
1 <= (E-min (L~ z)) .. z
by A6, FINSEQ_3:27;
then
(W-min (L~ z)) .. z > 1
by A1, Lm11, XXREAL_0:2;
then reconsider M = mid z,((W-min (L~ z)) .. z),(len z) as S-Sequence_in_R2 by A18, Th42;
A21:
len z in dom z
by FINSEQ_5:6;
then A22:
M /. 1 = z /. ((W-min (L~ z)) .. z)
by A15, Th12;
1 <= (W-min (L~ z)) .. z
by A15, FINSEQ_3:27;
then A23:
len M = ((len z) -' ((W-min (L~ z)) .. z)) + 1
by A17, JORDAN4:20;
A24:
M is_in_the_area_of z
by A15, A21, Th25, Th26;
A25:
M /. (len M) = z /. (len z)
by A15, A21, Th13;
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 A26:
1 < (E-min (L~ z)) .. z
by A1, A20, A7, XXREAL_0:1;
A27:
(W-max (L~ z)) .. z <= len z
by A12, FINSEQ_3:27;
then reconsider h = mid z,((W-max (L~ z)) .. z),((E-min (L~ z)) .. z) as S-Sequence_in_R2 by A4, A26, Th41;
A28: z /. ((W-max (L~ z)) .. z) =
z . ((W-max (L~ z)) .. z)
by A12, PARTFUN1:def 8
.=
W-max (L~ z)
by A11, FINSEQ_4:29
;
then
h /. 1 = W-max (L~ z)
by A12, A6, Th12;
then A29:
(h /. 1) `1 = W-bound (L~ z)
by EUCLID:56;
( h is_in_the_area_of z & h /. (len h) = z /. ((E-min (L~ z)) .. z) )
by A12, A6, Th13, Th25, Th26;
then A30:
( len h >= 2 & h is_a_h.c._for z )
by A8, A29, Def2, TOPREAL1:def 10;
W-max (L~ z) <> W-min (L~ z)
by Th62;
then
(W-min (L~ z)) .. z > (W-max (L~ z)) .. z
by A3, A28, A16, XXREAL_0:1;
then A31:
L~ M misses L~ h
by A17, A4, A26, Th54;
per cases
( SW-corner (L~ z) = W-min (L~ z) or SW-corner (L~ z) <> W-min (L~ z) )
;
suppose A32:
SW-corner (L~ z) = W-min (L~ z)
;
contradictionA33:
(M /. (len M)) `2 = N-bound (L~ z)
by A10, A25, EUCLID:56;
(M /. 1) `2 = S-bound (L~ z)
by A16, A22, A32, EUCLID:56;
then
M is_a_v.c._for z
by A24, A33, Def3;
hence
contradiction
by A30, A31, A23, A19, Th33;
verum end; suppose
SW-corner (L~ z) <> W-min (L~ z)
;
contradictionthen reconsider g =
<*(SW-corner (L~ z))*> ^ M as
S-Sequence_in_R2 by A15, A16, A21, Th71;
g /. 1
= SW-corner (L~ z)
by FINSEQ_5:16;
then A34:
(g /. 1) `2 = S-bound (L~ z)
by EUCLID:56;
(
len M in dom M &
len g = (len M) + (len <*(SW-corner (L~ z))*>) )
by FINSEQ_1:35, FINSEQ_5:6;
then
g /. (len g) = M /. (len M)
by FINSEQ_4:84;
then A35:
(g /. (len g)) `2 = N-bound (L~ z)
by A10, A25, EUCLID:56;
(LSeg (M /. 1),(SW-corner (L~ z))) /\ (L~ h) c= (LSeg (M /. 1),(SW-corner (L~ z))) /\ (L~ z)
by A13, A27, A20, A9, JORDAN4:47, XBOOLE_1:26;
then A36:
(LSeg (M /. 1),(SW-corner (L~ z))) /\ (L~ h) c= {(M /. 1)}
by A16, A22, PSCOMP_1:92;
(
L~ g = (L~ M) \/ (LSeg (SW-corner (L~ z)),(M /. 1)) &
M /. 1
in L~ M )
by A23, A19, JORDAN3:34, SPPOL_2:20;
then A37:
L~ g misses L~ h
by A31, A36, ZFMISC_1:149;
<*(SW-corner (L~ z))*> is_in_the_area_of z
by Th32;
then
g is_in_the_area_of z
by A24, Th28;
then
(
len g >= 2 &
g is_a_v.c._for z )
by A34, A35, Def3, TOPREAL1:def 10;
hence
contradiction
by A30, A37, Th33;
verum end;