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_right_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,(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 j1, i2 being Nat st front_right_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,(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 j1, i2 being Nat st front_right_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,(j1 + 1)] in Indices (Gauge (C,n)) )
assume that
A1:
f is_sequence_on Gauge (C,n)
and
A2:
len f > 1
; for j1, i2 being Nat st front_right_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,(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;
A4:
len (Gauge (C,n)) = width (Gauge (C,n))
by JORDAN8:def 1;
let j1, i2 be Nat; ( front_right_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,(j1 + 1)] in Indices (Gauge (C,n)) )
assume that
A5:
( front_right_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,(j1 + 1)] in Indices (Gauge (C,n))
A8:
i2 <= len (Gauge (C,n))
by A6, MATRIX_0:32;
A9:
j1 <= width (Gauge (C,n))
by A6, MATRIX_0:32;
A10:
now not j1 + 1 > len (Gauge (C,n))assume
j1 + 1
> len (Gauge (C,n))
;
contradictionthen A11:
(len (Gauge (C,n))) + 1
<= j1 + 1
by NAT_1:13;
j1 + 1
<= (len (Gauge (C,n))) + 1
by A9, A4, XREAL_1:6;
then
j1 + 1
= (len (Gauge (C,n))) + 1
by A11, XXREAL_0:1;
then
cell (
(Gauge (C,n)),
(i2 -' 1),
(len (Gauge (C,n))))
meets C
by A1, A5, A6, A7, A3, GOBRD13:39;
hence
contradiction
by A8, JORDAN8:15, NAT_D:44;
verum end;
A12:
1 <= j1 + 1
by NAT_1:11;
1 <= i2
by A6, MATRIX_0:32;
hence
[i2,(j1 + 1)] in Indices (Gauge (C,n))
by A8, A12, A4, A10, MATRIX_0:30; verum