let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for j, n, i being Element of NAT st j <= width (Gauge (C,n)) & cell ((Gauge (C,n)),i,j) c= BDD C holds
i <> len (Gauge (C,n))
let j, n, i be Element of NAT ; :: thesis: ( j <= width (Gauge (C,n)) & cell ((Gauge (C,n)),i,j) c= BDD C implies i <> len (Gauge (C,n)) )
assume that
A1: j <= width (Gauge (C,n)) and
A2: cell ((Gauge (C,n)),i,j) c= BDD C ; :: thesis: i <> len (Gauge (C,n))
A3: not cell ((Gauge (C,n)),(len (Gauge (C,n))),j) is empty by A1, JORDAN1A:45;
assume A4: i = len (Gauge (C,n)) ; :: thesis: contradiction
len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;
then cell ((Gauge (C,n)),(len (Gauge (C,n))),j) c= UBD C by A1, Th39;
hence contradiction by A2, A4, A3, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum