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 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) holds
ex k1 being Nat st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((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 Nat st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) holds
ex k1 being Nat st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
let i, j, k be Nat; ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) implies ex k1 being Nat st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((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,k) in L~ (Upper_Seq (C,n))
; ex k1 being Nat st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
set G = Gauge (C,n);
A7:
k >= 1
by A3, A4, XXREAL_0:2;
then A8:
[i,k] in Indices (Gauge (C,n))
by A1, A2, A5, MATRIX_0:30;
set X = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)));
A9:
(Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))
by RLTOPSP1:68;
then 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 A6, XBOOLE_0:def 4;
A10:
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets L~ (Upper_Seq (C,n))
by A6, A9, XBOOLE_0:3;
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)))));
A11:
j <= width (Gauge (C,n))
by A4, A5, XXREAL_0:2;
then
[i,j] in Indices (Gauge (C,n))
by A1, A2, A3, MATRIX_0:30;
then consider k1 being Nat such that
A12:
j <= k1
and
A13:
k1 <= k
and
A14:
((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, A10, A8, JORDAN1F:1, JORDAN1G: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))))))]|;
A15:
k1 <= width (Gauge (C,n))
by A5, A13, XXREAL_0:2;
((Gauge (C,n)) * (i,j)) `1 =
((Gauge (C,n)) * (i,1)) `1
by A1, A2, A3, A11, GOBOARD5:2
.=
((Gauge (C,n)) * (i,k)) `1
by A1, A2, A5, A7, GOBOARD5:2
;
then A16:
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) is vertical
by SPPOL_1:16;
take
k1
; ( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
thus
( j <= k1 & k1 <= k )
by A12, A13; (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))}
consider pp being object such that
A17:
pp in S-most X1
by XBOOLE_0:def 1;
A18:
1 <= k1
by A3, A12, XXREAL_0:2;
then A19:
((Gauge (C,n)) * (i,k1)) `1 = ((Gauge (C,n)) * (i,1)) `1
by A1, A2, A15, GOBOARD5:2;
then A20:
|[(((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 A14, EUCLID:53;
then A21:
((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, A12, A15, SPRECT_3:12;
A22:
((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, A11, A19, A20, GOBOARD5:2;
reconsider pp = pp as Point of (TOP-REAL 2) by A17;
A23:
pp in (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)))
by A17, XBOOLE_0:def 4;
then A24:
pp in L~ (Upper_Seq (C,n))
by XBOOLE_0:def 4;
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))))))]| `2 =
S-bound ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))
by A14, A20, SPRECT_1:44
.=
(S-min ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))) `2
by EUCLID:52
.=
pp `2
by A17, PSCOMP_1:55
;
pp in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))
by A23, XBOOLE_0:def 4;
then
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 A22, A16, SPPOL_1:41;
then A26:
|[(((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 A24, A25, TOPREAL3:6;
for x being object 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,j)))) /\ (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
object ;
( 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,j)))) /\ (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,j)))) /\ (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,j)))) /\ (L~ (Upper_Seq (C,n))) )proof
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:15;
A27:
(Gauge (C,n)) * (
i,
j)
in LSeg (
((Gauge (C,n)) * (i,j)),
((Gauge (C,n)) * (i,k)))
by RLTOPSP1:68;
A28:
((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, A7, A19, A20, GOBOARD5:2;
A29:
|[(((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, A13, A18, A20, SPRECT_3:12;
A30:
((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, A12, A15, A20, SPRECT_3:12;
((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, A11, A19, A20, GOBOARD5:2;
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))))))]| in LSeg (
((Gauge (C,n)) * (i,j)),
((Gauge (C,n)) * (i,k)))
by A28, A30, A29, GOBOARD7:7;
then A31:
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)))
c= LSeg (
((Gauge (C,n)) * (i,j)),
((Gauge (C,n)) * (i,k)))
by A27, TOPREAL1:6;
assume A32:
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,j)))) /\ (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) ;
A33:
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,j)))
by A32, XBOOLE_0:def 4;
then A34:
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 A21, TOPREAL1:4;
pp in L~ (Upper_Seq (C,n))
by A32, XBOOLE_0:def 4;
then
pp in EE
by A33, A31, XBOOLE_0:def 4;
then
proj2 . pp in E0
by FUNCT_2:35;
then A35:
pp `2 in E0
by PSCOMP_1:def 6;
E0 is
real-bounded
by RCOMP_1:10;
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 A14, A20, A35, SEQ_4:def 2;
then A36:
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 A34, 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 A22, A33, 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 A36, TOPREAL3:6;
verum
end;
assume A37:
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,j)))) /\ (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,j)))
by RLTOPSP1:68;
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,j)))) /\ (L~ (Upper_Seq (C,n)))
by A26, A37, XBOOLE_0:def 4;
verum
end;
hence
(LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))}
by A20, TARSKI:def 1; verum