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) & j <= width (Gauge C,n) & cell (Gauge C,n),i,j c= BDD C holds
i > 1
let i, n, j be Element of NAT ; :: thesis: ( i <= len (Gauge C,n) & j <= width (Gauge C,n) & cell (Gauge C,n),i,j c= BDD C implies i > 1 )
assume that
A1: i <= len (Gauge C,n) and
A2: j <= width (Gauge C,n) and
A3: cell (Gauge C,n),i,j c= BDD C and
A4: i <= 1 ; :: thesis: contradiction

per cases ( i = 0 or i = 1 ) by A4, NAT_1:26;

suppose i = 0 ; :: thesis: contradiction

hence contradiction by A2, A3, Lm3; :: thesis: verum

end;

suppose A5: i = 1 ; :: thesis: contradiction

A6: len (Gauge C,n) <> 0 by GOBOARD1:def 5;
then A7: 0 + 1 <= len (Gauge C,n) by NAT_1:14;
then A8: not cell (Gauge C,n),1,j is empty by A2, JORDAN1A:45;
A9: cell (Gauge C,n),0 ,j c= UBD C by A2, Th38;
A10: j < width (Gauge C,n) by A1, A2, A3, Lm6, XXREAL_0:1;
A11: 0 < len (Gauge C,n) by A6;
j <> 0 by A1, A3, Lm4;
then (cell (Gauge C,n),0 ,j) /\ (cell (Gauge C,n),(0 + 1),j) = LSeg ((Gauge C,n) * (0 + 1),j),((Gauge C,n) * (0 + 1),(j + 1)) by A10, A11, GOBOARD5:26, NAT_1:14;
then A12: cell (Gauge C,n),0 ,j meets cell (Gauge C,n),(0 + 1),j by XBOOLE_0:def 7;
UBD C is_outside_component_of C by JORDAN2C:73, JORDAN2C:76;
then A13: UBD C is_a_component_of C ` by JORDAN2C:def 4;
BDD C c= C ` by JORDAN2C:29;
then cell (Gauge C,n),1,j c= C ` by A3, A5, XBOOLE_1:1;
then cell (Gauge C,n),1,j c= UBD C by A2, A7, A9, A12, A13, GOBOARD9:6, JORDAN1A:46;
hence contradiction by A3, A5, A8, JORDAN2C:28, XBOOLE_1:68; :: thesis: verum

end;

end;