reserve i,j,k,k1,k2,i1,i2,j1,j2 for Nat,
  r,s for Real,
  x for set,
  f for non constant standard special_circular_sequence;

theorem Th3:
  for i,j st i<=len GoB f & j<=width GoB f holds Cl Down(Int cell(
GoB f,i,j),(L~f)`) is connected & Down(Int cell(GoB f,i,j),(L~f)`)=Int cell(GoB
  f,i,j)
proof
  let i,j;
  assume
A1: i<=len GoB f & j<=width GoB f;
  then
  Int cell(GoB f,i,j) is convex & Down(Int cell(GoB f,i,j),(L~f)`)=Int
  cell (GoB f,i,j) by Th1,GOBOARD9:17,XBOOLE_1:28;
  then Down(Int cell(GoB f,i,j),(L~f)`) is connected by CONNSP_1:23;
  hence Cl Down(Int cell(GoB f,i,j),(L~f)`) is connected by CONNSP_1:19;
  thus thesis by A1,Th1,XBOOLE_1:28;
end;
