let C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for n, m being Element of NAT st n <= m & n is_sufficiently_large_for C holds
m is_sufficiently_large_for C
let n, m be Element of NAT ; :: thesis: ( n <= m & n is_sufficiently_large_for C implies m is_sufficiently_large_for C )
assume that
A1: n <= m and
A2: n is_sufficiently_large_for C ; :: thesis: m is_sufficiently_large_for C
consider j being Element of NAT such that
A3: j < width (Gauge C,n) and
A4: cell (Gauge C,n),((X-SpanStart C,n) -' 1),j c= BDD C by A2, JORDAN1H:def 3;
set iin = X-SpanStart C,n;
set iim = X-SpanStart C,m;
n >= 1 by A2, JORDAN1H:60;
then A5: X-SpanStart C,m = ((2 |^ (m -' n)) * ((X-SpanStart C,n) - 2)) + 2 by A1, Th28;
A6: j > 1

proof

A7: (X-SpanStart C,n) -' 1 <= len (Gauge C,n) by JORDAN1H:59;
assume A8: j <= 1 ; :: thesis: contradiction

per cases ( j = 0 or j = 1 ) by A8, NAT_1:26;

suppose A9: j = 0 ; :: thesis: contradiction

width (Gauge C,n) >= 0 by NAT_1:2;
then A10: not cell (Gauge C,n),((X-SpanStart C,n) -' 1),0 is empty by A7, JORDAN1A:45;
cell (Gauge C,n),((X-SpanStart C,n) -' 1),0 c= UBD C by A7, JORDAN1A:70;
hence contradiction by A4, A9, A10, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

suppose A11: j = 1 ; :: thesis: contradiction

set i1 = X-SpanStart C,n;
A12: (X-SpanStart C,n) -' 1 <= X-SpanStart C,n by NAT_D:44;
X-SpanStart C,n < len (Gauge C,n) by JORDAN1H:58;
then A13: (X-SpanStart C,n) -' 1 < len (Gauge C,n) by A12, XXREAL_0:2;
BDD C c= C ` by JORDAN2C:29;
then A14: cell (Gauge C,n),((X-SpanStart C,n) -' 1),1 c= C ` by A4, A11, XBOOLE_1:1;
UBD C is_outside_component_of C by JORDAN2C:76;
then A15: UBD C is_a_component_of C ` by JORDAN2C:def 4;
width (Gauge C,n) <> 0 by GOBOARD1:def 5;
then A16: 0 + 1 <= width (Gauge C,n) by NAT_1:14;
then A17: not cell (Gauge C,n),((X-SpanStart C,n) -' 1),1 is empty by A7, JORDAN1A:45;
A18: 1 <= (X-SpanStart C,n) -' 1 by JORDAN1H:59;
0 < width (Gauge C,n) by A16, NAT_1:13;
then (cell (Gauge C,n),((X-SpanStart C,n) -' 1),0 ) /\ (cell (Gauge C,n),((X-SpanStart C,n) -' 1),(0 + 1)) = LSeg ((Gauge C,n) * ((X-SpanStart C,n) -' 1),(0 + 1)),((Gauge C,n) * (((X-SpanStart C,n) -' 1) + 1),(0 + 1)) by A13, A18, GOBOARD5:27;
then A19: cell (Gauge C,n),((X-SpanStart C,n) -' 1),0 meets cell (Gauge C,n),((X-SpanStart C,n) -' 1),(0 + 1) by XBOOLE_0:def 7;
cell (Gauge C,n),((X-SpanStart C,n) -' 1),0 c= UBD C by A7, JORDAN1A:70;
then cell (Gauge C,n),((X-SpanStart C,n) -' 1),1 c= UBD C by A16, A13, A19, A15, A14, GOBOARD9:6, JORDAN1A:46;
hence contradiction by A4, A11, A17, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

end;

end;

then ((2 |^ (m -' n)) * (j - 2)) + 2 > 1 by A1, A3, JORDAN1A:53;
then reconsider j1 = ((2 |^ (m -' n)) * (j - 2)) + 2 as Element of NAT by INT_1:16, XXREAL_0:2;
X-SpanStart C,n > 2 by JORDAN1H:58;
then A20: X-SpanStart C,n >= 2 + 1 by NAT_1:13;
A21: j + 1 < width (Gauge C,n)

proof

assume j + 1 >= width (Gauge C,n) ; :: thesis: contradiction
then A22: ( j + 1 > width (Gauge C,n) or j + 1 = width (Gauge C,n) ) by XXREAL_0:1;
A23: (X-SpanStart C,n) -' 1 <= len (Gauge C,n) by JORDAN1H:59;

per cases ( j = width (Gauge C,n) or j + 1 = width (Gauge C,n) ) by A3, A22, NAT_1:13;

suppose A24: j = width (Gauge C,n) ; :: thesis: contradiction

A25: cell (Gauge C,n),((X-SpanStart C,n) -' 1),(width (Gauge C,n)) c= UBD C by A23, JORDAN1A:71;
not cell (Gauge C,n),((X-SpanStart C,n) -' 1),j is empty by A23, A24, JORDAN1A:45;
hence contradiction by A4, A24, A25, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

suppose j + 1 = width (Gauge C,n) ; :: thesis: contradiction

then A26: cell (Gauge C,n),((X-SpanStart C,n) -' 1),((width (Gauge C,n)) -' 1) c= BDD C by A4, NAT_D:34;
BDD C c= C ` by JORDAN2C:29;
then A27: cell (Gauge C,n),((X-SpanStart C,n) -' 1),((width (Gauge C,n)) -' 1) c= C ` by A26, XBOOLE_1:1;
A28: cell (Gauge C,n),((X-SpanStart C,n) -' 1),(width (Gauge C,n)) c= UBD C by A23, JORDAN1A:71;
set i1 = X-SpanStart C,n;
A29: (X-SpanStart C,n) -' 1 <= X-SpanStart C,n by NAT_D:44;
X-SpanStart C,n < len (Gauge C,n) by JORDAN1H:58;
then A30: (X-SpanStart C,n) -' 1 < len (Gauge C,n) by A29, XXREAL_0:2;
UBD C is_outside_component_of C by JORDAN2C:76;
then A31: UBD C is_a_component_of C ` by JORDAN2C:def 4;
(width (Gauge C,n)) -' 1 <= width (Gauge C,n) by NAT_D:44;
then A32: not cell (Gauge C,n),((X-SpanStart C,n) -' 1),((width (Gauge C,n)) -' 1) is empty by A23, JORDAN1A:45;
A33: (width (Gauge C,n)) - 1 < width (Gauge C,n) by XREAL_1:148;
A34: 1 <= (X-SpanStart C,n) -' 1 by JORDAN1H:59;
A35: width (Gauge C,n) <> 0 by GOBOARD1:def 5;
then ((width (Gauge C,n)) -' 1) + 1 = width (Gauge C,n) by NAT_1:14, XREAL_1:237;
then (cell (Gauge C,n),((X-SpanStart C,n) -' 1),(width (Gauge C,n))) /\ (cell (Gauge C,n),((X-SpanStart C,n) -' 1),((width (Gauge C,n)) -' 1)) = LSeg ((Gauge C,n) * ((X-SpanStart C,n) -' 1),(width (Gauge C,n))),((Gauge C,n) * (((X-SpanStart C,n) -' 1) + 1),(width (Gauge C,n))) by A30, A33, A34, GOBOARD5:27;
then A36: cell (Gauge C,n),((X-SpanStart C,n) -' 1),(width (Gauge C,n)) meets cell (Gauge C,n),((X-SpanStart C,n) -' 1),((width (Gauge C,n)) -' 1) by XBOOLE_0:def 7;
(width (Gauge C,n)) -' 1 < width (Gauge C,n) by A35, A33, NAT_1:14, XREAL_1:235;
then cell (Gauge C,n),((X-SpanStart C,n) -' 1),((width (Gauge C,n)) -' 1) c= UBD C by A28, A30, A36, A31, A27, GOBOARD9:6, JORDAN1A:46;
hence contradiction by A26, A32, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

end;

end;

X-SpanStart C,n < len (Gauge C,n) by JORDAN1H:58;
then cell (Gauge C,m),((X-SpanStart C,m) -' 1),j1 c= cell (Gauge C,n),((X-SpanStart C,n) -' 1),j by A1, A5, A20, A6, A21, JORDAN1A:69;
then A37: cell (Gauge C,m),((X-SpanStart C,m) -' 1),j1 c= BDD C by A4, XBOOLE_1:1;
j1 < width (Gauge C,m) by A1, A3, A6, JORDAN1A:53;
hence m is_sufficiently_large_for C by A37, JORDAN1H:def 3; :: thesis: verum