reserve E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  i, j, m, n for Nat,
  p for Point of TOP-REAL 2;

theorem Th37:
  i <= len Gauge(C,n) & j <= width Gauge(C,n) & cell(Gauge(C,n),i,
  j) c= BDD C implies j > 1
proof
  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;
  per cases by A4,NAT_1:25;
  suppose
    j = 0;
    hence contradiction by A1,A3,Lm4;
  end;
  suppose
A5: j = 1;
    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: cell(Gauge(C,n),i,1) is non 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;
  end;
end;
