let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for i, j, n being Nat st i <= len (Gauge (C,n)) & j <= width (Gauge (C,n)) & cell ((Gauge (C,n)),i,j) c= BDD C holds
j > 1
let i, j, n be Nat; :: thesis: ( i <= len (Gauge (C,n)) & j <= width (Gauge (C,n)) & cell ((Gauge (C,n)),i,j) c= BDD C implies j > 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: j <= 1 ; :: thesis: contradiction

per cases ( j = 0 or j = 1 ) by A4, NAT_1:25;

suppose j = 0 ; :: thesis: contradiction

hence contradiction by A1, A3, Lm4; :: thesis: verum

end;

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

BDD C c= C ` by JORDAN2C:25;
then A6: cell ((Gauge (C,n)),i,1) c= C ` by A3, A5;
A7: i <> 0 by A2, A3, Lm3;
UBD C is_outside_component_of C by JORDAN2C:68;
then A8: UBD C is_a_component_of C ` by JORDAN2C:def 3;
A9: width (Gauge (C,n)) <> 0 by MATRIX_0:def 10;
then A10: 0 + 1 <= width (Gauge (C,n)) by NAT_1:14;
then A11: not cell ((Gauge (C,n)),i,1) is empty by A1, JORDAN1A:24;
i < len (Gauge (C,n)) by A1, A2, A3, Lm5, XXREAL_0:1;
then (cell ((Gauge (C,n)),i,0)) /\ (cell ((Gauge (C,n)),i,(0 + 1))) = LSeg (((Gauge (C,n)) * (i,(0 + 1))),((Gauge (C,n)) * ((i + 1),(0 + 1)))) by A9, A7, GOBOARD5:26, NAT_1:14;
then A12: cell ((Gauge (C,n)),i,0) meets cell ((Gauge (C,n)),i,(0 + 1)) by XBOOLE_0:def 7;
cell ((Gauge (C,n)),i,0) c= UBD C by A1, JORDAN1A:49;
then cell ((Gauge (C,n)),i,1) c= UBD C by A1, A10, A12, A8, A6, GOBOARD9:4, JORDAN1A:25;
hence contradiction by A3, A5, A11, JORDAN2C:24, XBOOLE_1:68; :: thesis: verum

end;

end;