let n be Element of NAT ; for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge C,n) & 1 <= j & j <= k & k <= width (Gauge C,n) & (Gauge C,n) * i,j in L~ (Lower_Seq C,n) & (Gauge C,n) * i,k in L~ (Upper_Seq C,n) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j1)} & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k1)} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge C,n) & 1 <= j & j <= k & k <= width (Gauge C,n) & (Gauge C,n) * i,j in L~ (Lower_Seq C,n) & (Gauge C,n) * i,k in L~ (Upper_Seq C,n) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j1)} & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k1)} )
let i, j, k be Element of NAT ; ( 1 <= i & i <= len (Gauge C,n) & 1 <= j & j <= k & k <= width (Gauge C,n) & (Gauge C,n) * i,j in L~ (Lower_Seq C,n) & (Gauge C,n) * i,k in L~ (Upper_Seq C,n) implies ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j1)} & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k1)} ) )
assume that
A1:
1 <= i
and
A2:
i <= len (Gauge C,n)
and
A3:
1 <= j
and
A4:
j <= k
and
A5:
k <= width (Gauge C,n)
and
A6:
(Gauge C,n) * i,j in L~ (Lower_Seq C,n)
and
A7:
(Gauge C,n) * i,k in L~ (Upper_Seq C,n)
; ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j1)} & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k1)} )
set G = Gauge C,n;
A8:
j <= width (Gauge C,n)
by A4, A5, XXREAL_0:2;
then A9:
[i,j] in Indices (Gauge C,n)
by A1, A2, A3, MATRIX_1:37;
set s = ((Gauge C,n) * i,1) `1 ;
set e = (Gauge C,n) * i,k;
set f = (Gauge C,n) * i,j;
set w1 = lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n))));
A10:
(Gauge C,n) * i,k in LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)
by RLTOPSP1:69;
then A11:
LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k) meets L~ (Upper_Seq C,n)
by A7, XBOOLE_0:3;
A12:
k >= 1
by A3, A4, XXREAL_0:2;
then
[i,k] in Indices (Gauge C,n)
by A1, A2, A5, MATRIX_1:37;
then consider k1 being Element of NAT such that
A13:
j <= k1
and
A14:
k1 <= k
and
A15:
((Gauge C,n) * i,k1) `2 = lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n))))
by A4, A11, A9, JORDAN1F:1, JORDAN1G:4;
A16:
k1 <= width (Gauge C,n)
by A5, A14, XXREAL_0:2;
A17:
(Gauge C,n) * i,j in LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k1)
by RLTOPSP1:69;
then A18:
LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k1) meets L~ (Lower_Seq C,n)
by A6, XBOOLE_0:3;
set X = (LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n));
reconsider X1 = (LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)) as non empty compact Subset of (TOP-REAL 2) by A6, A17, XBOOLE_0:def 4;
consider pp being set such that
A19:
pp in N-most X1
by XBOOLE_0:def 1;
reconsider pp = pp as Point of (TOP-REAL 2) by A19;
A20:
pp in (LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n))
by A19, XBOOLE_0:def 4;
then A21:
pp in L~ (Lower_Seq C,n)
by XBOOLE_0:def 4;
set p = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|;
set w2 = upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n))));
set q = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|;
A22:
pp in LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k1)
by A20, XBOOLE_0:def 4;
A23:
1 <= k1
by A3, A13, XXREAL_0:2;
then A24:
((Gauge C,n) * i,k1) `1 = ((Gauge C,n) * i,1) `1
by A1, A2, A16, GOBOARD5:3;
then A25:
|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| = (Gauge C,n) * i,k1
by A15, EUCLID:57;
[i,k1] in Indices (Gauge C,n)
by A1, A2, A23, A16, MATRIX_1:37;
then consider j1 being Element of NAT such that
A26:
j <= j1
and
A27:
j1 <= k1
and
A28:
((Gauge C,n) * i,j1) `2 = upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n))))
by A9, A13, A25, A18, JORDAN1F:2, JORDAN1G:5;
take
j1
; ex k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j1)} & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k1)} )
take
k1
; ( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j1)} & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k1)} )
thus
( j <= j1 & j1 <= k1 & k1 <= k )
by A14, A26, A27; ( (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j1)} & (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k1)} )
A29:
j1 <= width (Gauge C,n)
by A16, A27, XXREAL_0:2;
A30:
1 <= j1
by A3, A26, XXREAL_0:2;
then A31:
((Gauge C,n) * i,j1) `1 = ((Gauge C,n) * i,1) `1
by A1, A2, A29, GOBOARD5:3;
then A32:
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| = (Gauge C,n) * i,j1
by A28, EUCLID:57;
then A33:
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `2 <= |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `2
by A1, A2, A16, A25, A27, A30, SPRECT_3:24;
A34: |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `2 =
N-bound ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)))
by A25, A28, A32, SPRECT_1:50
.=
(N-min ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)))) `2
by EUCLID:56
.=
pp `2
by A19, PSCOMP_1:98
;
A35:
((Gauge C,n) * i,j) `1 = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `1
by A1, A2, A3, A8, A24, A25, GOBOARD5:3;
then
LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| is vertical
by SPPOL_1:37;
then
pp `1 = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `1
by A24, A25, A31, A32, A22, SPPOL_1:64;
then A36:
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| in L~ (Lower_Seq C,n)
by A21, A34, TOPREAL3:11;
for x being set holds
( x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)) iff x = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| )
proof
let x be
set ;
( x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)) iff x = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| )
thus
(
x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)) implies
x = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| )
( x = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| implies x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)) )proof
reconsider EE =
(LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)) as
compact Subset of
(TOP-REAL 2) ;
reconsider E0 =
proj2 .: EE as
compact Subset of
REAL by JCT_MISC:24;
A37:
|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| in LSeg ((Gauge C,n) * i,j),
|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|
by RLTOPSP1:69;
A38:
((Gauge C,n) * i,j) `2 <= |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `2
by A1, A2, A3, A26, A29, A32, SPRECT_3:24;
((Gauge C,n) * i,j) `1 = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `1
by A1, A2, A3, A8, A31, A32, GOBOARD5:3;
then
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| in LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,
((Gauge C,n) * i,j)
by A24, A25, A31, A32, A33, A38, GOBOARD7:8;
then A39:
LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| c= LSeg ((Gauge C,n) * i,j),
|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|
by A37, TOPREAL1:12;
assume A40:
x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n))
;
x = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|
then reconsider pp =
x as
Point of
(TOP-REAL 2) ;
A41:
pp in LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|
by A40, XBOOLE_0:def 4;
then A42:
pp `2 >= |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `2
by A33, TOPREAL1:10;
pp in L~ (Lower_Seq C,n)
by A40, XBOOLE_0:def 4;
then
pp in EE
by A41, A39, XBOOLE_0:def 4;
then
proj2 . pp in E0
by FUNCT_2:43;
then A43:
pp `2 in E0
by PSCOMP_1:def 29;
E0 is
bounded
by RCOMP_1:28;
then
E0 is
bounded_above
by XXREAL_2:def 11;
then
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `2 >= pp `2
by A28, A32, A43, SEQ_4:def 4;
then A44:
pp `2 = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `2
by A42, XXREAL_0:1;
pp `1 = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `1
by A24, A25, A31, A32, A41, GOBOARD7:5;
hence
x = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|
by A44, TOPREAL3:11;
verum
end;
assume A45:
x = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|
;
x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n))
then
x in LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|
by RLTOPSP1:69;
hence
x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n))
by A36, A45, XBOOLE_0:def 4;
verum
end;
hence
(LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j1)}
by A25, A32, TARSKI:def 1; (LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k1)}
set X = (LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n));
reconsider X1 = (LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)) as non empty compact Subset of (TOP-REAL 2) by A7, A10, XBOOLE_0:def 4;
consider pp being set such that
A46:
pp in S-most X1
by XBOOLE_0:def 1;
reconsider pp = pp as Point of (TOP-REAL 2) by A46;
A47:
pp in (LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n))
by A46, XBOOLE_0:def 4;
then A48:
pp in L~ (Upper_Seq C,n)
by XBOOLE_0:def 4;
((Gauge C,n) * i,j) `1 =
((Gauge C,n) * i,1) `1
by A1, A2, A3, A8, GOBOARD5:3
.=
((Gauge C,n) * i,k) `1
by A1, A2, A5, A12, GOBOARD5:3
;
then A49:
LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k) is vertical
by SPPOL_1:37;
pp in LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)
by A47, XBOOLE_0:def 4;
then A50:
pp `1 = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `1
by A35, A49, SPPOL_1:64;
|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `2 =
S-bound ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))
by A15, A25, SPRECT_1:49
.=
(S-min ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))) `2
by EUCLID:56
.=
pp `2
by A46, PSCOMP_1:118
;
then A51:
|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| in L~ (Upper_Seq C,n)
by A48, A50, TOPREAL3:11;
for x being set holds
( x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Upper_Seq C,n)) iff x = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| )
proof
let x be
set ;
( x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Upper_Seq C,n)) iff x = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| )
thus
(
x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Upper_Seq C,n)) implies
x = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| )
( x = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| implies x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Upper_Seq C,n)) )proof
A52:
|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `2 <= ((Gauge C,n) * i,k) `2
by A1, A2, A5, A14, A23, A25, SPRECT_3:24;
A53:
((Gauge C,n) * i,j) `2 <= |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `2
by A1, A2, A3, A13, A16, A25, SPRECT_3:24;
A54:
((Gauge C,n) * i,k) `1 = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `1
by A1, A2, A5, A12, A24, A25, GOBOARD5:3;
((Gauge C,n) * i,j) `1 = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `1
by A1, A2, A3, A8, A24, A25, GOBOARD5:3;
then A55:
|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| in LSeg ((Gauge C,n) * i,j),
((Gauge C,n) * i,k)
by A54, A53, A52, GOBOARD7:8;
A56:
((Gauge C,n) * i,k) `1 = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `1
by A1, A2, A5, A12, A31, A32, GOBOARD5:3;
j1 <= k
by A14, A27, XXREAL_0:2;
then A57:
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `2 <= ((Gauge C,n) * i,k) `2
by A1, A2, A5, A30, A32, SPRECT_3:24;
A58:
((Gauge C,n) * i,j) `2 <= |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `2
by A1, A2, A3, A26, A29, A32, SPRECT_3:24;
((Gauge C,n) * i,j) `1 = |[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| `1
by A1, A2, A3, A8, A31, A32, GOBOARD5:3;
then
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| in LSeg ((Gauge C,n) * i,j),
((Gauge C,n) * i,k)
by A56, A58, A57, GOBOARD7:8;
then A59:
LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]| c= LSeg ((Gauge C,n) * i,j),
((Gauge C,n) * i,k)
by A55, TOPREAL1:12;
reconsider EE =
(LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)) as
compact Subset of
(TOP-REAL 2) ;
reconsider E0 =
proj2 .: EE as
compact Subset of
REAL by JCT_MISC:24;
assume A60:
x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Upper_Seq C,n))
;
x = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|
then reconsider pp =
x as
Point of
(TOP-REAL 2) ;
A61:
pp in LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|
by A60, XBOOLE_0:def 4;
then A62:
pp `2 <= |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `2
by A33, TOPREAL1:10;
pp in L~ (Upper_Seq C,n)
by A60, XBOOLE_0:def 4;
then
pp in EE
by A61, A59, XBOOLE_0:def 4;
then
proj2 . pp in E0
by FUNCT_2:43;
then A63:
pp `2 in E0
by PSCOMP_1:def 29;
E0 is
bounded
by RCOMP_1:28;
then
E0 is
bounded_below
by XXREAL_2:def 11;
then
|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `2 <= pp `2
by A15, A25, A63, SEQ_4:def 5;
then A64:
pp `2 = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `2
by A62, XXREAL_0:1;
pp `1 = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]| `1
by A24, A25, A31, A32, A61, GOBOARD7:5;
hence
x = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|
by A64, TOPREAL3:11;
verum
end;
assume A65:
x = |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|
;
x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Upper_Seq C,n))
then
x in LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,
|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|
by RLTOPSP1:69;
hence
x in (LSeg |[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|,|[(((Gauge C,n) * i,1) `1 ),(upper_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),|[(((Gauge C,n) * i,1) `1 ),(lower_bound (proj2 .: ((LSeg ((Gauge C,n) * i,j),((Gauge C,n) * i,k)) /\ (L~ (Upper_Seq C,n)))))]|) /\ (L~ (Lower_Seq C,n)))))]|) /\ (L~ (Upper_Seq C,n))
by A51, A65, XBOOLE_0:def 4;
verum
end;
hence
(LSeg ((Gauge C,n) * i,j1),((Gauge C,n) * i,k1)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k1)}
by A25, A32, TARSKI:def 1; verum