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 Th21:
  1 <= j & j < width G & 1 <= m & m <= len G & 1 <= n & n <= width
  G & p in cell(G,len G,j) & p`1 = G*(m,n)`1 implies len G = m
proof
  assume that
A1: 1 <= j and
A2: j < width 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,len G,j) and
A8: p`1 = G*(m,n)`1;
A9: G*(m,1)`1 = G*(m,n)`1 by A3,A4,A5,A6,GOBOARD5:2;
A10: cell(G,len G,j) = { |[r,s]| where r, s is Real:
G*(len G,1)`1 <= r & G*
  (1,j)`2 <= s & s <= G*(1,j+1)`2 } by A1,A2,GOBRD11:29;
A11: 1 <= width G by A1,A2,XXREAL_0:2;
  consider r, s being Real such that
A12: p = |[r,s]| and
A13: G*(len G,1)`1 <= r and
  G*(1,j)`2 <= s and
  s <= G*(1,j+1)`2 by A7,A10;
  p`1 = r by A12,EUCLID:52;
  then len G <= m by A3,A8,A11,A9,A13,GOBOARD5:3;
  hence thesis by A4,XXREAL_0:1;
end;
