
theorem Th4:
  for G be Go-board for p be Point of TOP-REAL 2
 for i,j be Nat st 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G
  holds p in Int cell(G
,i,j) iff G*(i,j)`1 < p`1 & p`1 < G*(i+1,j)`1 & G*(i,j)`2 < p`2 & p`2 < G*(i,j+
  1)`2
proof
  let G be Go-board;
  let p be Point of TOP-REAL 2;
  let i,j be Nat;
  assume that
A1: 1 <= i and
A2: i+1 <= len G and
A3: 1 <= j and
A4: j+1 <= width G;
  set Z = {|[r,s]| where r,s is Real
    : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,
  j)`2 < s & s < G*(1,j+1)`2};
A5: j < width G by A4,NAT_1:13;
  i+1 >= 1 by NAT_1:11;
  then
A6: G*(i+1,1)`1 = G*(i+1,j)`1 by A2,A3,A5,GOBOARD5:2;
A7: i < len G by A2,NAT_1:13;
  then
A8: G*(1,j)`2 = G*(i,j)`2 by A1,A3,A5,GOBOARD5:1;
  j+1 >= 1 by NAT_1:11;
  then
A9: G*(1,j+1)`2 = G*(i,j+1)`2 by A1,A4,A7,GOBOARD5:1;
A10: G*(i,1)`1 = G*(i,j)`1 by A1,A3,A7,A5,GOBOARD5:2;
  thus p in Int cell(G,i,j) implies G*(i,j)`1 < p`1 & p`1 < G*(i+1,j)`1 & G*(i
  ,j)`2 < p`2 & p`2 < G*(i,j+1)`2
  proof
    assume p in Int cell(G,i,j);
    then p in Z by A1,A3,A7,A5,GOBOARD6:26;
    then
    ex r,s be Real st p = |[r,s]| & G*(i,1)`1 < r & r < G*(i +1,1)`1 &
    G*(1,j)`2 < s & s < G*(1,j+1)`2;
    hence thesis by A10,A6,A8,A9,EUCLID:52;
  end;
  assume that
A11: G*(i,j)`1 < p`1 and
A12: p`1 < G*(i+1,j)`1 and
A13: G*(i,j)`2 < p`2 and
A14: p`2 < G*(i,j+1)`2;
  p = |[p`1,p`2]| by EUCLID:53;
  then p in Z by A10,A6,A8,A9,A11,A12,A13,A14;
  hence thesis by A1,A3,A7,A5,GOBOARD6:26;
end;
