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 Th39:
  i <= len Gauge(C,n) & j <= width Gauge(C,n) & cell(Gauge(C,n),i,
  j) c= BDD C implies j+1 < width Gauge(C,n)
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;
A4: j < width Gauge(C,n) or j = width Gauge(C,n) by A2,XXREAL_0:1;
  assume j + 1 >= width Gauge(C,n);
  then
A5: j + 1 > width Gauge(C,n) or j + 1 = width Gauge(C,n) by XXREAL_0:1;
  per cases by A5,A4,NAT_1:13;
  suppose
    j = width Gauge(C,n);
    hence contradiction by A1,A3,Lm6;
  end;
  suppose
    j + 1 = width Gauge(C,n);
    then
A6: cell(Gauge(C,n),i,width Gauge(C,n)-'1) c= BDD C by A3,NAT_D:34;
    BDD C c= C` by JORDAN2C:25;
    then
A7: cell(Gauge(C,n),i,width Gauge(C,n)-'1) c= C` by A6;
A8: width Gauge(C,n) <> 0 by MATRIX_0:def 10;
    then
A9: width Gauge(C,n)-'1+1 = width Gauge(C,n) by NAT_1:14,XREAL_1:235;
    width Gauge(C,n)-'1 <= width Gauge(C,n) by NAT_D:44;
    then
A10: cell(Gauge(C,n),i,width Gauge(C,n)-'1) is non empty by A1,JORDAN1A:24;
A11: cell(Gauge(C,n),i,width Gauge(C,n)) c= UBD C by A1,JORDAN1A:50;
    UBD C is_outside_component_of C by JORDAN2C:68;
    then
A12: UBD C is_a_component_of C` by JORDAN2C:def 3;
A13: i <> 0 by A2,A3,Lm3;
A14: width Gauge(C,n)-1 < width Gauge(C,n) by XREAL_1:146;
    i < len Gauge(C,n) by A1,A2,A3,Lm5,XXREAL_0:1;
    then
    cell(Gauge(C,n),i,width Gauge(C,n)) /\ cell(Gauge(C,n),i,width Gauge(
C,n)-'1) = LSeg(Gauge(C,n)*(i,width Gauge(C,n)), Gauge(C,n)*(i+1,width Gauge(C,
    n))) by A14,A9,A13,GOBOARD5:26,NAT_1:14;
    then
A15: cell(Gauge(C,n),i,width Gauge(C,n)) meets cell(Gauge(C,n),i,width
    Gauge(C,n)-'1) by XBOOLE_0:def 7;
    width Gauge(C,n)-'1 < width Gauge(C,n) by A8,A14,NAT_1:14,XREAL_1:233;
    then cell(Gauge(C,n),i,width Gauge(C,n)-'1) c= UBD C by A1,A11,A15,A12,A7,
GOBOARD9:4,JORDAN1A:25;
    hence contradiction by A6,A10,JORDAN2C:24,XBOOLE_1:68;
  end;
end;
