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 Th23:
  1 <= i & i < len G & 1 <= m & m <= len G & 1 <= n & n <= width G
  & p in cell(G,i,width G) & p`2 = G*(m,n)`2 implies width G = n
proof
  assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 <= m and
A4: m <= len G and
A5: 1 <= n and
A6: n <= width G and
A7: p in cell(G,i,width G) and
A8: p`2 = G*(m,n)`2;
A9: G*(1,n)`2 = G*(m,n)`2 by A3,A4,A5,A6,GOBOARD5:1;
A10: cell(G,i,width G) = { |[r,s]| where r, s is Real:
  G*(i,1)`1 <= r & r <=
  G*(i+1,1)`1 & G*(1,width G)`2 <= s } by A1,A2,GOBRD11:31;
A11: 1 <= len G by A1,A2,XXREAL_0:2;
  consider r, s being Real such that
A12: p = |[r,s]| and
  G*(i,1)`1 <= r and
  r <= G*(i+1,1)`1 and
A13: G*(1,width G)`2 <= s by A7,A10;
  p`2 = s by A12,EUCLID:52;
  then width G <= n by A5,A8,A11,A9,A13,GOBOARD5:4;
  hence thesis by A6,XXREAL_0:1;
end;
