let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for i, n, j being Element of NAT st i <= len (Gauge (C,n)) & cell ((Gauge (C,n)),i,j) c= BDD C holds
j <> 0
let i, n, j be Element of NAT ; :: thesis: ( i <= len (Gauge (C,n)) & cell ((Gauge (C,n)),i,j) c= BDD C implies j <> 0 )
assume that
A1: i <= len (Gauge (C,n)) and
A2: cell ((Gauge (C,n)),i,j) c= BDD C and
A3: j = 0 ; :: thesis: contradiction
A4: cell ((Gauge (C,n)),i,0) c= UBD C by A1, JORDAN1A:70;
0 <= width (Gauge (C,n)) ;
then not cell ((Gauge (C,n)),i,0) is empty by A1, JORDAN1A:45;
hence contradiction by A2, A3, A4, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum