reserve i,j,k,n,m for Nat;
reserve p,q for Point of TOP-REAL 2;
reserve G for Go-board;
reserve C for Subset of TOP-REAL 2;

theorem
  for f being non constant standard special_circular_sequence st 1 <= i
& i < len GoB f & 1 <= j & j < width GoB f holds Int cell(GoB f,i,j) misses L~
  SpStSeq L~f
proof
  let f be non constant standard special_circular_sequence such that
A1: 1 <= i and
A2: i < len GoB f and
A3: 1 <= j and
A4: j < width GoB f;
A5: i+1 <= len GoB f by A2,NAT_1:13;
  set G = GoB f;
A6: Int cell(G,i,j) = { |[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 }
   by A1,A2,A3,A4,GOBOARD6:26;
A7: N-bound L~SpStSeq L~f = N-bound L~f by SPRECT_1:60;
A8: 1 <= width GoB f by GOBRD11:34;
  then
A9: <*(GoB f)*(i,1)*> is_in_the_area_of f by A1,A2,Th49;
  1 <= i+1 by A1,NAT_1:13;
  then
A10: <*(GoB f)*(i+1,1)*> is_in_the_area_of f by A8,A5,Th49;
  assume Int cell(GoB f,i,j) meets L~SpStSeq L~f;
  then consider x being object such that
A11: x in Int cell(GoB f,i,j) and
A12: x in L~SpStSeq L~f by XBOOLE_0:3;
A13: W-bound L~SpStSeq L~f = W-bound L~f by SPRECT_1:58;
A14: 1 <= len GoB f by GOBRD11:34;
  then
A15: <*(GoB f)*(1,j)*> is_in_the_area_of f by A3,A4,Th49;
A16: j+1 <= width GoB f by A4,NAT_1:13;
  1 <= j+1 by A3,NAT_1:13;
  then
A17: <*(GoB f)*(1,j+1)*> is_in_the_area_of f by A14,A16,Th49;
A18: L~SpStSeq L~f = { p: p`1 = W-bound L~SpStSeq L~f & p`2 <= N-bound L~
SpStSeq L~f & p`2 >= S-bound L~SpStSeq L~f or p`1 <= E-bound L~SpStSeq L~f & p
`1 >= W-bound L~SpStSeq L~f & p`2 = N-bound L~SpStSeq L~f or p`1 <= E-bound L~
SpStSeq L~f & p`1 >= W-bound L~SpStSeq L~f & p`2 = S-bound L~SpStSeq L~f or p`1
  = E-bound L~SpStSeq L~f & p`2 <= N-bound L~SpStSeq L~f & p`2 >= S-bound L~
  SpStSeq L~f} by Th35;
A19: E-bound L~SpStSeq L~f = E-bound L~f by SPRECT_1:61;
  consider p such that
A20: p = x and
A21: p`1 = W-bound L~SpStSeq L~f & p`2 <= N-bound L~SpStSeq L~f & p`2 >=
  S-bound L~SpStSeq L~f or p`1 <= E-bound L~SpStSeq L~f & p`1 >= W-bound L~
SpStSeq L~f & p`2 = N-bound L~SpStSeq L~f or p`1 <= E-bound L~SpStSeq L~f & p`1
  >= W-bound L~SpStSeq L~f & p`2 = S-bound L~SpStSeq L~f or p`1 = E-bound L~
  SpStSeq L~f & p`2 <= N-bound L~SpStSeq L~f & p`2 >= S-bound L~SpStSeq L~f by
A12,A18;
A22: S-bound L~SpStSeq L~f = S-bound L~f by SPRECT_1:59;
  consider r,s being Real such that
A23: x = |[r,s]| and
A24: G*(i,1)`1 < r and
A25: r < G*(i+1,1)`1 and
A26: G*(1,j)`2 < s and
A27: s < G*(1,j+1)`2 by A6,A11;
A28: p`1 = r by A23,A20,EUCLID:52;
A29: p`2 = s by A23,A20,EUCLID:52;
  per cases by A21;
  suppose
A30: p`1 = W-bound L~SpStSeq L~f;
A31: 1 in dom<*G*(i,1)*> by FINSEQ_5:6;
    <*G*(i,1)*>/.1 = G*(i,1) by FINSEQ_4:16;
    hence contradiction by A24,A9,A28,A13,A30,A31;
  end;
  suppose
A32: p`2 = N-bound L~SpStSeq L~f;
A33: 1 in dom<*G*(1,j+1)*> by FINSEQ_5:6;
    <*G*(1,j+1)*>/.1 = G*(1,j+1) by FINSEQ_4:16;
    hence contradiction by A27,A17,A29,A7,A32,A33;
  end;
  suppose that
A34: p`2 = S-bound L~SpStSeq L~f;
A35: 1 in dom<*G*(1,j)*> by FINSEQ_5:6;
    <*G*(1,j)*>/.1 = G*(1,j) by FINSEQ_4:16;
    hence contradiction by A26,A15,A29,A22,A34,A35;
  end;
  suppose that
A36: p`1 = E-bound L~SpStSeq L~f;
A37: 1 in dom<*G*(i+1,1)*> by FINSEQ_5:6;
    <*G*(i+1,1)*>/.1 = G*(i+1,1) by FINSEQ_4:16;
    hence contradiction by A25,A10,A28,A19,A36,A37;
  end;
end;
