let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for i, j, n being Nat st j <= width (Gauge (C,n)) & cell ((Gauge (C,n)),i,j) c= BDD C holds
i <> 0
let i, j, n be 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, Th35;
0 <= len (Gauge (C,n)) ;
then not cell ((Gauge (C,n)),0,j) is empty by A1, JORDAN1A:24;
hence contradiction by A2, A3, A4, JORDAN2C:24, XBOOLE_1:68; :: thesis: verum