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 Th6:
  for i1,j1,i2,j2 st i1<=len GoB f & j1<=width GoB f & i2<=len GoB
f & j2<=width GoB f & i1,j1,i2,j2 are_adjacent holds Int cell(GoB f,i1,j1) c=
LeftComp f \/ RightComp f iff Int cell(GoB f,i2,j2) c= LeftComp f \/ RightComp
  f
proof
  let i1,j1,i2,j2;
  assume that
A1: i1<=len GoB f & j1<=width GoB f and
A2: i2<=len GoB f and
A3: j2<=width GoB f and
A4: i1,j1,i2,j2 are_adjacent;
A5: i1,i2 are_adjacent & j1=j2 or i1=i2 & j1,j2 are_adjacent by A4,
GOBRD10:def 2;
  now
    per cases by A5,GOBRD10:def 1;
    case
      (i2=i1+1 or i1=i2+1)& j1=j2;
      hence thesis by A1,A2,Lm5;
    end;
    case
      i1=i2 &(j2=j1+1 or j1=j2+1);
      hence thesis by A1,A3,Lm6;
    end;
  end;
  hence thesis;
end;
