let E be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for j, n being Element of NAT st j <= len (Gauge (E,n)) holds
cell ((Gauge (E,n)),(len (Gauge (E,n))),j) c= UBD E
let j, n be Element of NAT ; :: thesis: ( j <= len (Gauge (E,n)) implies cell ((Gauge (E,n)),(len (Gauge (E,n))),j) c= UBD E )
A1: not E ` is empty by JORDAN2C:73, JORDAN2C:74;
assume A2: j <= len (Gauge (E,n)) ; :: thesis: cell ((Gauge (E,n)),(len (Gauge (E,n))),j) c= UBD E
then cell ((Gauge (E,n)),(len (Gauge (E,n))),j) misses E by JORDAN8:19;
then A3: cell ((Gauge (E,n)),(len (Gauge (E,n))),j) c= E ` by SUBSET_1:43;
A4: width (Gauge (E,n)) = len (Gauge (E,n)) by JORDAN8:def 1;
then A5: not cell ((Gauge (E,n)),(len (Gauge (E,n))),j) is empty by A2, JORDAN1A:45;
cell ((Gauge (E,n)),(len (Gauge (E,n))),j) is connected by A4, A2, JORDAN1A:46;
then consider W being Subset of (TOP-REAL 2) such that
A6: W is_a_component_of E ` and
A7: cell ((Gauge (E,n)),(len (Gauge (E,n))),j) c= W by A3, A1, A5, GOBOARD9:5;
not W is Bounded by A4, A2, A7, Th37, JORDAN2C:16;
then W is_outside_component_of E by A6, JORDAN2C:def 4;
then W c= UBD E by JORDAN2C:27;
hence cell ((Gauge (E,n)),(len (Gauge (E,n))),j) c= UBD E by A7, XBOOLE_1:1; :: thesis: verum