let n be Nat; for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being 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~ (Lower_Seq (C,n)) holds
ex j1 being Nat st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); for i, j, k being 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~ (Lower_Seq (C,n)) holds
ex j1 being Nat st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
let i, j, k be Nat; ( 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) implies ex j1 being Nat st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,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~ (Lower_Seq (C,n))
; ex j1 being Nat st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
set G = Gauge (C,n);
A7:
k >= 1
by A1, A2, XXREAL_0:2;
then A8:
[k,i] in Indices (Gauge (C,n))
by A3, A4, A5, MATRIX_0:30;
set X = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)));
A9:
(Gauge (C,n)) * (j,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))
by RLTOPSP1:68;
then 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 A6, XBOOLE_0:def 4;
A10:
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets L~ (Lower_Seq (C,n))
by A6, A9, XBOOLE_0:3;
set s = ((Gauge (C,n)) * (1,i)) `2 ;
set e = (Gauge (C,n)) * (k,i);
set f = (Gauge (C,n)) * (j,i);
set w2 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))));
A11:
len (Gauge (C,n)) = width (Gauge (C,n))
by JORDAN8:def 1;
then A12:
j <= width (Gauge (C,n))
by A2, A3, XXREAL_0:2;
then
[j,i] in Indices (Gauge (C,n))
by A1, A4, A5, A11, MATRIX_0:30;
then consider j1 being Nat such that
A13:
j <= j1
and
A14:
j1 <= k
and
A15:
((Gauge (C,n)) * (j1,i)) `1 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))
by A2, A10, A8, JORDAN1F:4, JORDAN1G:5;
set q = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|;
A16:
1 <= j1
by A1, A13, XXREAL_0:2;
take
j1
; ( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
thus
( j <= j1 & j1 <= k )
by A13, A14; (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))}
consider pp being object such that
A17:
pp in E-most X1
by XBOOLE_0:def 1;
reconsider pp = pp as Point of (TOP-REAL 2) by A17;
A18:
pp in (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))
by A17, XBOOLE_0:def 4;
then A19:
pp in L~ (Lower_Seq (C,n))
by XBOOLE_0:def 4;
A20:
j1 <= width (Gauge (C,n))
by A3, A11, A14, XXREAL_0:2;
then A21:
((Gauge (C,n)) * (j1,i)) `2 = ((Gauge (C,n)) * (1,i)) `2
by A4, A5, A11, A16, GOBOARD5:1;
then A22:
|[(upper_bound (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)) * (j1,i)
by A15, EUCLID:53;
then A23:
|[(upper_bound (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, A14, A16, SPRECT_3:13;
A24:
((Gauge (C,n)) * (k,i)) `2 = ((Gauge (C,n)) * (1,i)) `2
by A3, A4, A5, A7, GOBOARD5:1;
then
((Gauge (C,n)) * (j,i)) `2 = ((Gauge (C,n)) * (k,i)) `2
by A1, A4, A5, A11, A12, GOBOARD5:1;
then A25:
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) is horizontal
by SPPOL_1:15;
A26: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 =
E-bound ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))))
by A15, A22, SPRECT_1:46
.=
(E-min ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))))) `1
by EUCLID:52
.=
pp `1
by A17, PSCOMP_1:47
;
pp in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))
by A18, XBOOLE_0:def 4;
then
pp `2 = |[(upper_bound (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 A24, A21, A22, A25, SPPOL_1:40;
then A27:
|[(upper_bound (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 A19, A26, TOPREAL3:6;
for x being object holds
( x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(upper_bound (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
object ;
( x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(upper_bound (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 (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) implies
x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
( x = |[(upper_bound (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 (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) )proof
A28:
((Gauge (C,n)) * (j,i)) `1 <= |[(upper_bound (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, A11, A13, A20, A22, SPRECT_3:13;
((Gauge (C,n)) * (j,i)) `2 = |[(upper_bound (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, A11, A12, A21, A22, GOBOARD5:1;
then A29:
|[(upper_bound (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)) * (k,i)),
((Gauge (C,n)) * (j,i)))
by A24, A21, A22, A23, A28, GOBOARD7:8;
(Gauge (C,n)) * (
k,
i)
in LSeg (
((Gauge (C,n)) * (j,i)),
((Gauge (C,n)) * (k,i)))
by RLTOPSP1:68;
then A30:
LSeg (
((Gauge (C,n)) * (k,i)),
|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)
c= LSeg (
((Gauge (C,n)) * (j,i)),
((Gauge (C,n)) * (k,i)))
by A29, TOPREAL1:6;
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) ;
reconsider E0 =
proj1 .: EE as
compact Subset of
REAL by Th4;
assume A31:
x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))
;
x = |[(upper_bound (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) ;
A32:
pp in LSeg (
((Gauge (C,n)) * (k,i)),
|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)
by A31, XBOOLE_0:def 4;
then A33:
pp `1 >= |[(upper_bound (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 A23, TOPREAL1:3;
pp in L~ (Lower_Seq (C,n))
by A31, XBOOLE_0:def 4;
then
pp in EE
by A32, A30, XBOOLE_0:def 4;
then
proj1 . pp in E0
by FUNCT_2:35;
then A34:
pp `1 in E0
by PSCOMP_1:def 5;
E0 is
real-bounded
by RCOMP_1:10;
then
E0 is
bounded_above
by XXREAL_2:def 11;
then
|[(upper_bound (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 A15, A22, A34, SEQ_4:def 1;
then A35:
pp `1 = |[(upper_bound (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 A33, XXREAL_0:1;
pp `2 = |[(upper_bound (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 A24, A21, A22, A32, GOBOARD7:6;
hence
x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|
by A35, TOPREAL3:6;
verum
end;
assume A36:
x = |[(upper_bound (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 (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))
then
x in LSeg (
((Gauge (C,n)) * (k,i)),
|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)
by RLTOPSP1:68;
hence
x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))
by A27, A36, XBOOLE_0:def 4;
verum
end;
hence
(LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))}
by A22, TARSKI:def 1; verum