let n be Nat; for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Nat st 1 <= i & i < len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); for i being Nat st 1 <= i & i < len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n))
let i be Nat; ( 1 <= i & i < len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) )
set wi = width (Gauge (C,n));
assume that
A1:
( 1 <= i & i < len (Gauge (C,n)) )
and
A2:
(Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n))
; contradiction
consider i2 being Nat such that
A3:
i2 in dom (Lower_Seq (C,n))
and
A4:
(Lower_Seq (C,n)) . i2 = (Gauge (C,n)) * (i,(width (Gauge (C,n))))
by A2, FINSEQ_2:10;
reconsider i2 = i2 as Nat ;
A5:
( 1 <= i2 & i2 <= len (Lower_Seq (C,n)) )
by A3, FINSEQ_3:25;
3 <= len (Upper_Seq (C,n))
by JORDAN1E:15;
then A6:
2 <= len (Upper_Seq (C,n))
by XXREAL_0:2;
set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
set i1 = (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n));
A7:
len (Gauge (C,n)) = width (Gauge (C,n))
by JORDAN8:def 1;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n))
by SPRECT_2:46;
then A8:
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n)))
by FINSEQ_6:92;
L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))
by REVROT_1:33;
then A9:
( (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) & (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) )
by A8, SPRECT_5:40, SPRECT_5:41;
A10:
( W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) & rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) )
by FINSEQ_6:90, SPRECT_2:43, SPRECT_2:46;
(W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n))
by Th30;
then
(S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n))
by Th29;
then A11:
(S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n))
by Th28, XXREAL_0:2;
( (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1 & (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) )
by Th25, Th27;
then A12:
(S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) > 1
by Th26, XXREAL_0:2;
then A13:
(S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) in dom (Lower_Seq (C,n))
by A11, FINSEQ_3:25;
( Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) & S-max (L~ (Cage (C,n))) in rng (Cage (C,n)) )
by Th18, SPRECT_2:42;
then A14:
S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n))
by A10, A9, FINSEQ_5:46, XXREAL_0:2;
then A15:
(Lower_Seq (C,n)) /. ((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n)))
by FINSEQ_5:38;
A16:
( (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) in NAT & i2 in NAT )
by ORDINAL1:def 12;
A17:
(S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <> i2
proof
assume
(S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = i2
;
contradiction
then
(Gauge (C,n)) * (
i,
(width (Gauge (C,n))))
= S-max (L~ (Cage (C,n)))
by A4, A13, A15, PARTFUN1:def 6;
then
((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = S-bound (L~ (Cage (C,n)))
by EUCLID:52;
then
N-bound (L~ (Cage (C,n))) = S-bound (L~ (Cage (C,n)))
by A1, A7, JORDAN1A:70;
hence
contradiction
by SPRECT_1:16;
verum
end;
then
mid ((Lower_Seq (C,n)),((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))),i2) is being_S-Seq
by A12, A11, A5, JORDAN3:6, A16;
then reconsider h = mid ((Lower_Seq (C,n)),((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))),i2) as one-to-one special FinSequence of (TOP-REAL 2) ;
A18: (h /. 1) `2 =
((Lower_Seq (C,n)) /. ((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)))) `2
by A3, A13, SPRECT_2:8
.=
(S-max (L~ (Cage (C,n)))) `2
by A14, FINSEQ_5:38
.=
S-bound (L~ (Cage (C,n)))
by EUCLID:52
;
len h >= 1
by A3, A13, SPRECT_2:5;
then
len h > 1
by A3, A13, A17, SPRECT_2:6, XXREAL_0:1;
then A19:
1 + 1 <= len h
by NAT_1:13;
A20:
h is_in_the_area_of Cage (C,n)
by A3, A13, JORDAN1E:18, SPRECT_2:22;
(h /. (len h)) `2 =
((Lower_Seq (C,n)) /. i2) `2
by A3, A13, SPRECT_2:9
.=
((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2
by A3, A4, PARTFUN1:def 6
.=
N-bound (L~ (Cage (C,n)))
by A1, A7, JORDAN1A:70
;
then
h is_a_v.c._for Cage (C,n)
by A20, A18, SPRECT_2:def 3;
then
L~ (Upper_Seq (C,n)) meets L~ h
by A6, A19, Th40, SPRECT_2:29;
then consider x being object such that
A21:
x in L~ (Upper_Seq (C,n))
and
A22:
x in L~ h
by XBOOLE_0:3;
L~ (mid ((Lower_Seq (C,n)),((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))),i2)) c= L~ (Lower_Seq (C,n))
by A12, A11, A5, JORDAN4:35;
then
x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n)))
by A21, A22, XBOOLE_0:def 4;
then A23:
x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))}
by JORDAN1E:16;
4 <= len (Gauge (C,n))
by JORDAN8:10;
then A24:
1 <= len (Gauge (C,n))
by XXREAL_0:2;
per cases
( x = E-max (L~ (Cage (C,n))) or x = W-min (L~ (Cage (C,n))) )
by A23, TARSKI:def 2;
suppose
x = E-max (L~ (Cage (C,n)))
;
contradictionthen
x = (Lower_Seq (C,n)) /. 1
by JORDAN1F:6;
then
i2 = 1
by A12, A11, A5, A22, Th37;
then
(Lower_Seq (C,n)) /. 1
= (Gauge (C,n)) * (
i,
(width (Gauge (C,n))))
by A3, A4, PARTFUN1:def 6;
then
E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (
i,
(width (Gauge (C,n))))
by JORDAN1F:6;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 =
E-bound (L~ (Cage (C,n)))
by EUCLID:52
.=
((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))) `1
by A7, A24, JORDAN1A:71
;
hence
contradiction
by A1, A7, A24, GOBOARD5:3;
verum end; suppose
x = W-min (L~ (Cage (C,n)))
;
contradictionthen
x = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))
by JORDAN1F:8;
then
i2 = len (Lower_Seq (C,n))
by A12, A11, A5, A22, Th38;
then
(Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Gauge (C,n)) * (
i,
(width (Gauge (C,n))))
by A3, A4, PARTFUN1:def 6;
then A25:
W-min (L~ (Cage (C,n))) = (Gauge (C,n)) * (
i,
(width (Gauge (C,n))))
by JORDAN1F:8;
(NW-corner (L~ (Cage (C,n)))) `2 >= (W-max (L~ (Cage (C,n)))) `2
by PSCOMP_1:30;
then
(NW-corner (L~ (Cage (C,n)))) `2 > (W-min (L~ (Cage (C,n)))) `2
by SPRECT_2:57, XXREAL_0:2;
then
N-bound (L~ (Cage (C,n))) > ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2
by A25, EUCLID:52;
hence
contradiction
by A1, A7, JORDAN1A:70;
verum end; end;