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 <> 0
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 <> 0 )
assume that
A1: j <= width (Gauge C,n) and
A2: cell (Gauge C,n),i,j c= BDD C and
A3: i = 0 ; :: thesis: contradiction
A4: cell (Gauge C,n),0 ,j c= UBD C by A1, Th38;
0 <= len (Gauge C,n) ;
then not cell (Gauge C,n),0 ,j is empty by A1, JORDAN1A:45;
hence contradiction by A2, A3, A4, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum