let n be Element of NAT ; :: thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) st n is_sufficiently_large_for C holds
n >= 1
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( n is_sufficiently_large_for C implies n >= 1 )
A1: 2 |^ 0 = 1 by NEWTON:9;
assume n is_sufficiently_large_for C ; :: thesis: n >= 1
then consider j being Element of NAT such that
A2: j < width (Gauge C,n) and
A3: cell (Gauge C,n),((X-SpanStart C,n) -' 1),j c= BDD C by Def3;
assume n < 1 ; :: thesis: contradiction
then A4: n = 0 by NAT_1:14;
A5: j > 1

proof

A6: (X-SpanStart C,n) -' 1 <= len (Gauge C,n) by Th59;
assume A7: j <= 1 ; :: thesis: contradiction

per cases ( j = 0 or j = 1 ) by A7, NAT_1:26;

suppose A8: j = 0 ; :: thesis: contradiction

0 <= width (Gauge C,n) ;
then A9: not cell (Gauge C,n),((X-SpanStart C,n) -' 1),0 is empty by A6, JORDAN1A:45;
cell (Gauge C,n),((X-SpanStart C,n) -' 1),0 c= UBD C by A6, JORDAN1A:70;
hence contradiction by A3, A8, A9, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

suppose A10: j = 1 ; :: thesis: contradiction

set i1 = X-SpanStart C,n;
UBD C is_outside_component_of C by JORDAN2C:73, JORDAN2C:76;
then A11: UBD C is_a_component_of C ` by JORDAN2C:def 4;
( X-SpanStart C,n < len (Gauge C,n) & (X-SpanStart C,n) -' 1 <= X-SpanStart C,n ) by Th58, NAT_D:44;
then A12: (X-SpanStart C,n) -' 1 < len (Gauge C,n) by XXREAL_0:2;
BDD C c= C ` by JORDAN2C:29;
then A13: cell (Gauge C,n),((X-SpanStart C,n) -' 1),1 c= C ` by A3, A10, XBOOLE_1:1;
A14: width (Gauge C,n) <> 0 by GOBOARD1:def 5;
then A15: 0 + 1 <= width (Gauge C,n) by NAT_1:14;
then A16: not cell (Gauge C,n),((X-SpanStart C,n) -' 1),1 is empty by A6, JORDAN1A:45;
1 <= (X-SpanStart C,n) -' 1 by Th59;
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 A14, A12, GOBOARD5:27;
then A17: 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 A6, JORDAN1A:70;
then cell (Gauge C,n),((X-SpanStart C,n) -' 1),1 c= UBD C by A15, A12, A17, A11, A13, GOBOARD9:6, JORDAN1A:46;
hence contradiction by A3, A10, A16, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

end;

end;

A18: width (Gauge C,n) = (2 |^ n) + 3 by JORDAN1A:49;
then A19: j <= 3 + 1 by A2, A4, NEWTON:9;
A20: j + 1 < width (Gauge C,n)

proof

assume j + 1 >= width (Gauge C,n) ; :: thesis: contradiction
then A21: ( j + 1 > width (Gauge C,n) or j + 1 = width (Gauge C,n) ) by XXREAL_0:1;
A22: (X-SpanStart C,n) -' 1 <= len (Gauge C,n) by Th59;

per cases ( j = width (Gauge C,n) or j + 1 = width (Gauge C,n) ) by A2, A21, NAT_1:13;

suppose A23: j = width (Gauge C,n) ; :: thesis: contradiction

( not cell (Gauge C,n),((X-SpanStart C,n) -' 1),(width (Gauge C,n)) is empty & cell (Gauge C,n),((X-SpanStart C,n) -' 1),(width (Gauge C,n)) c= UBD C ) by A22, JORDAN1A:45, JORDAN1A:71;
hence contradiction by A3, A23, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

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

then A24: cell (Gauge C,n),((X-SpanStart C,n) -' 1),((width (Gauge C,n)) -' 1) c= BDD C by A3, NAT_D:34;
BDD C c= C ` by JORDAN2C:29;
then A25: cell (Gauge C,n),((X-SpanStart C,n) -' 1),((width (Gauge C,n)) -' 1) c= C ` by A24, XBOOLE_1:1;
set i1 = X-SpanStart C,n;
A26: (width (Gauge C,n)) - 1 < width (Gauge C,n) by XREAL_1:148;
UBD C is_outside_component_of C by JORDAN2C:73, JORDAN2C:76;
then A27: 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 A28: not cell (Gauge C,n),((X-SpanStart C,n) -' 1),((width (Gauge C,n)) -' 1) is empty by A22, JORDAN1A:45;
A29: cell (Gauge C,n),((X-SpanStart C,n) -' 1),(width (Gauge C,n)) c= UBD C by A22, JORDAN1A:71;
A30: 1 <= (X-SpanStart C,n) -' 1 by Th59;
( X-SpanStart C,n < len (Gauge C,n) & (X-SpanStart C,n) -' 1 <= X-SpanStart C,n ) by Th58, NAT_D:44;
then A31: (X-SpanStart C,n) -' 1 < len (Gauge C,n) by XXREAL_0:2;
A32: 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 A31, A26, A30, GOBOARD5:27;
then A33: 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 A32, A26, 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 A29, A31, A33, A27, A25, GOBOARD9:6, JORDAN1A:46;
hence contradiction by A24, A28, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

end;

end;

per cases ( j = 0 or j = 1 or j = 2 or j = 3 or j = 4 ) by A19, NAT_1:29;

suppose ( j = 0 or j = 1 ) ; :: thesis: contradiction

hence contradiction by A5; :: thesis: verum

end;

suppose A34: j = 2 ; :: thesis: contradiction

A35: X-SpanStart C,0 = 1 + 2 by A1, NAT_2:10;
then (X-SpanStart C,0 ) -' 1 = (X-SpanStart C,0 ) - 1 by NAT_D:39
.= 2 by A35 ;
hence contradiction by A3, A4, A34, JORDAN1B:19; :: thesis: verum

end;

suppose ( j = 3 or j = 4 ) ; :: thesis: contradiction

hence contradiction by A18, A20, A1, A4; :: thesis: verum

end;

end;