let i, n be Element of NAT ; :: thesis: for E, C being compact non horizontal non vertical Subset of (TOP-REAL 2) st i <= len (Gauge (E,n)) holds
cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E
let E, C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( i <= len (Gauge (E,n)) implies cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E )
assume A1: i <= len (Gauge (E,n)) ; :: thesis: cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E
width (Gauge (E,n)) = len (Gauge (E,n)) by JORDAN8:def 1;
then cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) misses E by A1, JORDAN8:15;
then A2: cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= E ` by SUBSET_1:23;
A3: not E ` is empty by JORDAN2C:66;
( cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) is connected & not cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) is empty ) by A1, Th45, Th46;
then consider W being Subset of (TOP-REAL 2) such that
A4: W is_a_component_of E ` and
A5: cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= W by A2, A3, GOBOARD9:3;
not W is Bounded by A1, A5, Th48, JORDAN2C:12;
then W is_outside_component_of E by A4, JORDAN2C:def 3;
then W c= UBD E by JORDAN2C:23;
hence cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E by A5, XBOOLE_1:1; :: thesis: verum