let i, n be Element of NAT ; :: thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) st i <= len (Gauge C,n) holds
cell (Gauge C,n),i,0 c= UBD C
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( i <= len (Gauge C,n) implies cell (Gauge C,n),i,0 c= UBD C )
A1: not C ` is empty by JORDAN2C:74;
assume A2: i <= len (Gauge C,n) ; :: thesis: cell (Gauge C,n),i,0 c= UBD C
then cell (Gauge C,n),i,0 misses C by JORDAN8:20;
then A3: cell (Gauge C,n),i,0 c= C ` by SUBSET_1:43;
0 <= width (Gauge C,n) ;
then ( cell (Gauge C,n),i,0 is connected & not cell (Gauge C,n),i,0 is empty ) by A2, Th45, Th46;
then consider W being Subset of (TOP-REAL 2) such that
A4: W is_a_component_of C ` and
A5: cell (Gauge C,n),i,0 c= W by A3, A1, GOBOARD9:5;
not W is Bounded by A2, A5, Th47, JORDAN2C:16;
then W is_outside_component_of C by A4, JORDAN2C:def 4;
then W c= UBD C by JORDAN2C:27;
hence cell (Gauge C,n),i,0 c= UBD C by A5, XBOOLE_1:1; :: thesis: verum