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 Th17:
  E c= cell(Gauge(E,0),2,2)
proof
  set G = Gauge(E,0);
  let x be object;
A1: len G = width G by JORDAN8:def 1;
  assume
A2: x in E;
  then reconsider x as Point of TOP-REAL 2;
A3: 4 <= len G by JORDAN8:10;
  then
A4: 1 < len G by XXREAL_0:2;
  then G*(1,2)`2 = S-bound E by JORDAN8:13;
  then
A5: G*(1,2)`2 <= x`2 by A2,PSCOMP_1:24;
  2 < len G by A3,XXREAL_0:2;
  then
A6: cell(G,2,2) = { |[p,q]| where p, q is Real:
   G*(2,1)`1 <= p & p <= G*(2+
  1,1)`1 & G*(1,2)`2 <= q & q <= G*(1,2+1)`2 } by A1,GOBRD11:32;
  G*(2,1)`1 = W-bound E by A4,JORDAN8:11;
  then
A7: G*(2,1)`1 <= x`1 by A2,PSCOMP_1:24;
A8: len G-'1 = 3 by Lm1;
  then G*(2+1,1)`1 = E-bound E by A4,JORDAN8:12;
  then
A9: x`1 <= G*(2+1,1)`1 by A2,PSCOMP_1:24;
  G*(1,2+1)`2 = N-bound E by A8,A4,JORDAN8:14;
  then
A10: x`2 <= G*(1,2+1)`2 by A2,PSCOMP_1:24;
  x = |[x`1,x`2]| by EUCLID:53;
  hence thesis by A7,A9,A5,A10,A6;
end;
