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 <= j & j <= k & k <= len (Gauge C,n) & 1 <= i & i <= width (Gauge C,n) & (Gauge C,n) * j,i in L~ (Upper_Seq C,n) & (Gauge C,n) * k,i in L~ (Lower_Seq C,n) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * j1,i)} & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k1,i)} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge C,n) & 1 <= i & i <= width (Gauge C,n) & (Gauge C,n) * j,i in L~ (Upper_Seq C,n) & (Gauge C,n) * k,i in L~ (Lower_Seq C,n) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * j1,i)} & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k1,i)} )
let i, j, k be Element of NAT ; ( 1 <= j & j <= k & k <= len (Gauge C,n) & 1 <= i & i <= width (Gauge C,n) & (Gauge C,n) * j,i in L~ (Upper_Seq C,n) & (Gauge C,n) * k,i in L~ (Lower_Seq C,n) implies ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * j1,i)} & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k1,i)} ) )
assume that
A1:
1 <= j
and
A2:
j <= k
and
A3:
k <= len (Gauge C,n)
and
A4:
1 <= i
and
A5:
i <= width (Gauge C,n)
and
A6:
(Gauge C,n) * j,i in L~ (Upper_Seq C,n)
and
A7:
(Gauge C,n) * k,i in L~ (Lower_Seq C,n)
; ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * j1,i)} & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k1,i)} )
set G = Gauge C,n;
A8:
len (Gauge C,n) = width (Gauge C,n)
by JORDAN8:def 1;
then A9:
j <= width (Gauge C,n)
by A2, A3, XXREAL_0:2;
then A10:
[j,i] in Indices (Gauge C,n)
by A1, A4, A5, A8, MATRIX_1:37;
set s = ((Gauge C,n) * 1,i) `2 ;
set e = (Gauge C,n) * k,i;
set f = (Gauge C,n) * j,i;
set w1 = inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))));
A11:
(Gauge C,n) * k,i in LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)
by RLTOPSP1:69;
then A12:
LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) meets L~ (Lower_Seq C,n)
by A7, XBOOLE_0:3;
A13:
k >= 1
by A1, A2, XXREAL_0:2;
then
[k,i] in Indices (Gauge C,n)
by A3, A4, A5, MATRIX_1:37;
then consider k1 being Element of NAT such that
A14:
j <= k1
and
A15:
k1 <= k
and
A16:
((Gauge C,n) * k1,i) `1 = inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))
by A2, A12, A10, JORDAN1F:3, JORDAN1G:5;
A17:
k1 <= width (Gauge C,n)
by A3, A8, A15, XXREAL_0:2;
set p = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|;
set w2 = sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))));
set q = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|;
A18:
(Gauge C,n) * j,i in LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k1,i)
by RLTOPSP1:69;
then A19:
LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k1,i) meets L~ (Upper_Seq C,n)
by A6, XBOOLE_0:3;
A20:
1 <= k1
by A1, A14, XXREAL_0:2;
then A21:
((Gauge C,n) * k1,i) `2 = ((Gauge C,n) * 1,i) `2
by A4, A5, A8, A17, GOBOARD5:2;
then A22:
|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| = (Gauge C,n) * k1,i
by A16, EUCLID:57;
((Gauge C,n) * j,i) `2 =
((Gauge C,n) * 1,i) `2
by A1, A4, A5, A8, A9, GOBOARD5:2
.=
((Gauge C,n) * k,i) `2
by A3, A4, A5, A13, GOBOARD5:2
;
then A23:
LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) is horizontal
by SPPOL_1:36;
set X = (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n));
reconsider X1 = (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)) as non empty compact Subset of (TOP-REAL 2) by A6, A18, PSCOMP_1:64, XBOOLE_0:def 4;
consider pp being set such that
A24:
pp in E-most X1
by XBOOLE_0:def 1;
[k1,i] in Indices (Gauge C,n)
by A4, A5, A8, A20, A17, MATRIX_1:37;
then consider j1 being Element of NAT such that
A25:
j <= j1
and
A26:
j1 <= k1
and
A27:
((Gauge C,n) * j1,i) `1 = sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))
by A10, A14, A22, A19, JORDAN1F:4, JORDAN1G:4;
A28:
j1 <= width (Gauge C,n)
by A17, A26, XXREAL_0:2;
reconsider pp = pp as Point of (TOP-REAL 2) by A24;
A29:
pp in (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n))
by A24, XBOOLE_0:def 4;
then A30:
pp in L~ (Upper_Seq C,n)
by XBOOLE_0:def 4;
take
j1
; ex k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * j1,i)} & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k1,i)} )
take
k1
; ( j <= j1 & j1 <= k1 & k1 <= k & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * j1,i)} & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k1,i)} )
thus
( j <= j1 & j1 <= k1 & k1 <= k )
by A15, A25, A26; ( (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * j1,i)} & (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k1,i)} )
A31:
pp in LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k1,i)
by A29, XBOOLE_0:def 4;
A32:
1 <= j1
by A1, A25, XXREAL_0:2;
then A33:
((Gauge C,n) * j1,i) `2 = ((Gauge C,n) * 1,i) `2
by A4, A5, A8, A28, GOBOARD5:2;
then A34:
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| = (Gauge C,n) * j1,i
by A27, EUCLID:57;
then A35:
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1 <= |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1
by A4, A5, A8, A17, A22, A26, A32, SPRECT_3:25;
A36: |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1 =
E-bound ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)))
by A22, A27, A34, SPRECT_1:51
.=
(E-min ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)))) `1
by EUCLID:56
.=
pp `1
by A24, PSCOMP_1:108
;
A37:
((Gauge C,n) * j,i) `2 = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A1, A4, A5, A8, A9, A21, A22, GOBOARD5:2;
then
LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| is horizontal
by SPPOL_1:36;
then
pp `2 = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A21, A22, A33, A34, A31, SPPOL_1:63;
then A38:
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| in L~ (Upper_Seq C,n)
by A30, A36, TOPREAL3:11;
for x being set holds
( x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n)) iff x = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| )
proof
let x be
set ;
( x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n)) iff x = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| )
thus
(
x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n)) implies
x = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| )
( x = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| implies x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n)) )proof
reconsider EE =
(LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n)) as
compact Subset of
(TOP-REAL 2) by PSCOMP_1:64;
assume A39:
x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))
;
x = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
then reconsider pp =
x as
Point of
(TOP-REAL 2) ;
A40:
pp in LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
by A39, XBOOLE_0:def 4;
then A41:
pp `1 >= |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1
by A35, TOPREAL1:9;
A42:
((Gauge C,n) * j,i) `1 <= |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1
by A1, A4, A5, A8, A25, A28, A34, SPRECT_3:25;
reconsider E0 =
proj1 .: EE as
compact Subset of
REAL by Th4;
A43:
|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| in LSeg ((Gauge C,n) * j,i),
|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
by RLTOPSP1:69;
((Gauge C,n) * j,i) `2 = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A1, A4, A5, A8, A9, A33, A34, GOBOARD5:2;
then
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| in LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,
((Gauge C,n) * j,i)
by A21, A22, A33, A34, A35, A42, GOBOARD7:9;
then A44:
LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| c= LSeg ((Gauge C,n) * j,i),
|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
by A43, TOPREAL1:12;
pp in L~ (Upper_Seq C,n)
by A39, XBOOLE_0:def 4;
then
pp in EE
by A40, A44, XBOOLE_0:def 4;
then
proj1 . pp in E0
by FUNCT_2:43;
then A45:
pp `1 in E0
by PSCOMP_1:def 28;
E0 is
bounded
by RCOMP_1:28;
then
E0 is
bounded_above
by XXREAL_2:def 11;
then
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1 >= pp `1
by A27, A34, A45, SEQ_4:def 4;
then A46:
pp `1 = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1
by A41, XXREAL_0:1;
pp `2 = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A21, A22, A33, A34, A40, GOBOARD7:6;
hence
x = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
by A46, TOPREAL3:11;
verum
end;
assume A47:
x = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
;
x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))
then
x in LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
by RLTOPSP1:69;
hence
x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))
by A38, A47, XBOOLE_0:def 4;
verum
end;
hence
(LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * j1,i)}
by A22, A34, TARSKI:def 1; (LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k1,i)}
set X = (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n));
reconsider X1 = (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) as non empty compact Subset of (TOP-REAL 2) by A7, A11, PSCOMP_1:64, XBOOLE_0:def 4;
consider pp being set such that
A48:
pp in W-most X1
by XBOOLE_0:def 1;
reconsider pp = pp as Point of (TOP-REAL 2) by A48;
A49:
pp in (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))
by A48, XBOOLE_0:def 4;
then A50:
pp in L~ (Lower_Seq C,n)
by XBOOLE_0:def 4;
pp in LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)
by A49, XBOOLE_0:def 4;
then A51:
pp `2 = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A37, A23, SPPOL_1:63;
|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1 =
W-bound ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)))
by A16, A22, SPRECT_1:48
.=
(W-min ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)))) `1
by EUCLID:56
.=
pp `1
by A48, PSCOMP_1:88
;
then A52:
|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| in L~ (Lower_Seq C,n)
by A50, A51, TOPREAL3:11;
for x being set holds
( x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Lower_Seq C,n)) iff x = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| )
proof
let x be
set ;
( x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Lower_Seq C,n)) iff x = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| )
thus
(
x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Lower_Seq C,n)) implies
x = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| )
( x = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| implies x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Lower_Seq C,n)) )proof
j1 <= k
by A15, A26, XXREAL_0:2;
then A53:
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1 <= ((Gauge C,n) * k,i) `1
by A3, A4, A5, A32, A34, SPRECT_3:25;
A54:
((Gauge C,n) * k,i) `2 = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A3, A4, A5, A13, A21, A22, GOBOARD5:2;
A55:
((Gauge C,n) * j,i) `1 <= |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1
by A1, A4, A5, A8, A14, A17, A22, SPRECT_3:25;
A56:
((Gauge C,n) * j,i) `1 <= |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1
by A1, A4, A5, A8, A25, A28, A34, SPRECT_3:25;
A57:
|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1 <= ((Gauge C,n) * k,i) `1
by A3, A4, A5, A15, A20, A22, SPRECT_3:25;
((Gauge C,n) * j,i) `2 = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A1, A4, A5, A8, A9, A21, A22, GOBOARD5:2;
then A58:
|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| in LSeg ((Gauge C,n) * j,i),
((Gauge C,n) * k,i)
by A54, A55, A57, GOBOARD7:9;
A59:
((Gauge C,n) * k,i) `2 = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A3, A4, A5, A13, A33, A34, GOBOARD5:2;
((Gauge C,n) * j,i) `2 = |[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A1, A4, A5, A8, A9, A33, A34, GOBOARD5:2;
then
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| in LSeg ((Gauge C,n) * j,i),
((Gauge C,n) * k,i)
by A59, A56, A53, GOBOARD7:9;
then A60:
LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| c= LSeg ((Gauge C,n) * j,i),
((Gauge C,n) * k,i)
by A58, TOPREAL1:12;
reconsider EE =
(LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) as
compact Subset of
(TOP-REAL 2) by PSCOMP_1:64;
reconsider E0 =
proj1 .: EE as
compact Subset of
REAL by Th4;
assume A61:
x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Lower_Seq C,n))
;
x = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
then reconsider pp =
x as
Point of
(TOP-REAL 2) ;
A62:
pp in LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
by A61, XBOOLE_0:def 4;
then A63:
pp `1 <= |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1
by A35, TOPREAL1:9;
pp in L~ (Lower_Seq C,n)
by A61, XBOOLE_0:def 4;
then
pp in EE
by A62, A60, XBOOLE_0:def 4;
then
proj1 . pp in E0
by FUNCT_2:43;
then A64:
pp `1 in E0
by PSCOMP_1:def 28;
E0 is
bounded
by RCOMP_1:28;
then
E0 is
bounded_below
by XXREAL_2:def 11;
then
|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1 <= pp `1
by A16, A22, A64, SEQ_4:def 5;
then A65:
pp `1 = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `1
by A63, XXREAL_0:1;
pp `2 = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]| `2
by A21, A22, A33, A34, A62, GOBOARD7:6;
hence
x = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
by A65, TOPREAL3:11;
verum
end;
assume A66:
x = |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
;
x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Lower_Seq C,n))
then
x in LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,
|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|
by RLTOPSP1:69;
hence
x in (LSeg |[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|,|[(sup (proj1 .: ((LSeg ((Gauge C,n) * j,i),|[(inf (proj1 .: ((LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Upper_Seq C,n))))),(((Gauge C,n) * 1,i) `2 )]|) /\ (L~ (Lower_Seq C,n))
by A52, A66, XBOOLE_0:def 4;
verum
end;
hence
(LSeg ((Gauge C,n) * j1,i),((Gauge C,n) * k1,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k1,i)}
by A22, A34, TARSKI:def 1; verum