let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); for n being Nat
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on Gauge (C,n) & len f > 1 holds
for i1, j1 being Nat st front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [i1,j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * (i1,j1) & [(i1 + 1),j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * ((i1 + 1),j1) holds
[(i1 + 1),(j1 -' 1)] in Indices (Gauge (C,n))
let n be Nat; for f being FinSequence of (TOP-REAL 2) st f is_sequence_on Gauge (C,n) & len f > 1 holds
for i1, j1 being Nat st front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [i1,j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * (i1,j1) & [(i1 + 1),j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * ((i1 + 1),j1) holds
[(i1 + 1),(j1 -' 1)] in Indices (Gauge (C,n))
set G = Gauge (C,n);
let f be FinSequence of (TOP-REAL 2); ( f is_sequence_on Gauge (C,n) & len f > 1 implies for i1, j1 being Nat st front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [i1,j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * (i1,j1) & [(i1 + 1),j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * ((i1 + 1),j1) holds
[(i1 + 1),(j1 -' 1)] in Indices (Gauge (C,n)) )
assume that
A1:
f is_sequence_on Gauge (C,n)
and
A2:
len f > 1
; for i1, j1 being Nat st front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [i1,j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * (i1,j1) & [(i1 + 1),j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * ((i1 + 1),j1) holds
[(i1 + 1),(j1 -' 1)] in Indices (Gauge (C,n))
A3:
( 1 <= (len f) -' 1 & ((len f) -' 1) + 1 = len f )
by A2, NAT_D:49, XREAL_1:235;
let i1, j1 be Nat; ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [i1,j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * (i1,j1) & [(i1 + 1),j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * ((i1 + 1),j1) implies [(i1 + 1),(j1 -' 1)] in Indices (Gauge (C,n)) )
assume that
A4:
( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [i1,j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * (i1,j1) )
and
A5:
[(i1 + 1),j1] in Indices (Gauge (C,n))
and
A6:
f /. (len f) = (Gauge (C,n)) * ((i1 + 1),j1)
; [(i1 + 1),(j1 -' 1)] in Indices (Gauge (C,n))
A7:
i1 + 1 <= len (Gauge (C,n))
by A5, MATRIX_0:32;
A8:
1 <= j1
by A5, MATRIX_0:32;
A9:
now not j1 -' 1 < 1assume
j1 -' 1
< 1
;
contradictionthen
j1 <= 1
by NAT_1:14, NAT_D:36;
then
j1 = 1
by A8, XXREAL_0:1;
then
cell (
(Gauge (C,n)),
(i1 + 1),
(1 -' 1))
meets C
by A1, A4, A5, A6, A3, GOBRD13:37;
then
cell (
(Gauge (C,n)),
(i1 + 1),
0)
meets C
by XREAL_1:232;
hence
contradiction
by A7, JORDAN8:17;
verum end;
A10:
j1 -' 1 <= j1
by NAT_D:35;
j1 <= width (Gauge (C,n))
by A5, MATRIX_0:32;
then A11:
j1 -' 1 <= width (Gauge (C,n))
by A10, XXREAL_0:2;
1 <= i1 + 1
by A5, MATRIX_0:32;
hence
[(i1 + 1),(j1 -' 1)] in Indices (Gauge (C,n))
by A7, A11, A9, MATRIX_0:30; verum