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 j1, i2 being Nat st front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [(i2 + 1),j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * ((i2 + 1),j1) & [i2,j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i2,j1) holds
[(i2 -' 1),j1] 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 j1, i2 being Nat st front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [(i2 + 1),j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * ((i2 + 1),j1) & [i2,j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i2,j1) holds
[(i2 -' 1),j1] in Indices (Gauge (C,n))
set G = Gauge (C,n);
A1:
len (Gauge (C,n)) = width (Gauge (C,n))
by JORDAN8:def 1;
let f be FinSequence of (TOP-REAL 2); ( f is_sequence_on Gauge (C,n) & len f > 1 implies for j1, i2 being Nat st front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [(i2 + 1),j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * ((i2 + 1),j1) & [i2,j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i2,j1) holds
[(i2 -' 1),j1] in Indices (Gauge (C,n)) )
assume that
A2:
f is_sequence_on Gauge (C,n)
and
A3:
len f > 1
; for j1, i2 being Nat st front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [(i2 + 1),j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * ((i2 + 1),j1) & [i2,j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i2,j1) holds
[(i2 -' 1),j1] in Indices (Gauge (C,n))
A4:
( 1 <= (len f) -' 1 & ((len f) -' 1) + 1 = len f )
by A3, NAT_D:49, XREAL_1:235;
let j1, i2 be Nat; ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [(i2 + 1),j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * ((i2 + 1),j1) & [i2,j1] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i2,j1) implies [(i2 -' 1),j1] in Indices (Gauge (C,n)) )
assume that
A5:
( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C & [(i2 + 1),j1] in Indices (Gauge (C,n)) & f /. ((len f) -' 1) = (Gauge (C,n)) * ((i2 + 1),j1) )
and
A6:
[i2,j1] in Indices (Gauge (C,n))
and
A7:
f /. (len f) = (Gauge (C,n)) * (i2,j1)
; [(i2 -' 1),j1] in Indices (Gauge (C,n))
A8:
j1 <= width (Gauge (C,n))
by A6, MATRIX_0:32;
A9:
1 <= i2
by A6, MATRIX_0:32;
A10:
now not i2 -' 1 < 1assume
i2 -' 1
< 1
;
contradictionthen
i2 <= 1
by NAT_1:14, NAT_D:36;
then
i2 = 1
by A9, XXREAL_0:1;
then
cell (
(Gauge (C,n)),
(1 -' 1),
(j1 -' 1))
meets C
by A2, A5, A6, A7, A4, GOBRD13:38;
then
cell (
(Gauge (C,n)),
0,
(j1 -' 1))
meets C
by XREAL_1:232;
hence
contradiction
by A1, A8, JORDAN8:18, NAT_D:44;
verum end;
i2 <= len (Gauge (C,n))
by A6, MATRIX_0:32;
then A11:
i2 -' 1 <= len (Gauge (C,n))
by NAT_D:44;
1 <= j1
by A6, MATRIX_0:32;
hence
[(i2 -' 1),j1] in Indices (Gauge (C,n))
by A8, A11, A10, MATRIX_0:30; verum