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 Th37:
  for f being rectangular special_circular_sequence holds LeftComp
f = {p : not(W-bound L~f <= p`1 & p`1 <= E-bound L~f & S-bound L~f <= p`2 & p`2
  <= N-bound L~f)} & RightComp f = {q : W-bound L~f < q`1 & q`1 < E-bound L~f &
  S-bound L~f < q`2 & q`2 < N-bound L~f}
proof
  let f be rectangular special_circular_sequence;
  defpred U[Element of TOP-REAL 2] means not(W-bound L~f <= $1`1 & $1`1 <=
  E-bound L~f & S-bound L~f <= $1`2 & $1`2 <= N-bound L~f);
  defpred V[Element of TOP-REAL 2] means W-bound L~f < $1`1 & $1`1 < E-bound
  L~f & S-bound L~f < $1`2 & $1`2 < N-bound L~f;
  defpred W[Element of TOP-REAL 2] means $1`1 = W-bound L~f & $1`2 <= N-bound
L~f & $1`2 >= S-bound L~f or $1`1 <= E-bound L~f & $1`1 >= W-bound L~f & $1`2 =
  N-bound L~f or $1`1 <= E-bound L~f & $1`1 >= W-bound L~f & $1`2 = S-bound L~f
  or $1`1 = E-bound L~f & $1`2 <= N-bound L~f & $1`2 >= S-bound L~f;
  set LC = {p : U[p] }, RC = {q : V[q] }, Lf = {p : W[p] };
A1: S-bound L~f < N-bound L~f by SPRECT_1:32;
A2: Lf is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
A3: RC is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
A4: L~f = Lf by Th35;
  LC is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
  then reconsider Lc9=LC,Rc9=RC,Lf as Subset of TOP-REAL 2 by A3,A2;
  reconsider Lf as Subset of TOP-REAL 2;
  reconsider Lc9, Rc9 as Subset of TOP-REAL 2;
A5: W-bound L~f < E-bound L~f by SPRECT_1:31;
  then reconsider Lc=Lc9, Rc=Rc9 as Subset of (TOP-REAL 2)|Lf` by A1,JORDAN1:39
,41;
  reconsider Lc, Rc as Subset of (TOP-REAL 2)|Lf`;
A6: 1/2*(S-bound L~f) + 1/2*(S-bound L~f) = (S-bound L~f);
  Rc = Rc9;
  then Lc is a_component by A5,A1,JORDAN1:36;
  then
A7: Lc9 is_a_component_of Lf` by CONNSP_1:def 6;
  set p = 1/2*((GoB f)*(1,width GoB f)+(GoB f)*(1+1,width GoB f))+|[0,1]|, q =
  1/2*((GoB f)*(1,width GoB f)+(GoB f)*(1+1,width GoB f));
A8: 1+1 <= len f by GOBOARD7:34,XXREAL_0:2;
A9: p`2 = q`2+|[0,1]|`2 by TOPREAL3:2
    .= q`2+1 by EUCLID:52;
A10: GoB f = (f/.4,f/.1)][(f/.3,f/.2) by Th36;
  then
A11: 1+1 = width GoB f by MATRIX_0:47;
  then q`2 = (1/2*((GoB f)*(1,width GoB f)+f/.2))`2 by A10,MATRIX_0:50
    .= (1/2*(f/.1+f/.2))`2 by A10,A11,MATRIX_0:50
    .= 1/2*(f/.1+f/.2)`2 by TOPREAL3:4
    .= 1/2*((f/.1)`2+(f/.2)`2) by TOPREAL3:2
    .= 1/2*((N-min L~f)`2+(f/.2)`2) by SPRECT_1:83
    .= 1/2*((N-min L~f)`2+(N-max L~f)`2) by SPRECT_1:84
    .= 1/2*(N-bound L~f+(N-max L~f)`2) by EUCLID:52
    .= 1/2*(N-bound L~f+N-bound L~f) by EUCLID:52
    .= N-bound L~f;
  then p`2 > 0 + N-bound L~f by A9,XREAL_1:8;
  then
A12: p in LC;
A13: 1+1 = len GoB f by A10,MATRIX_0:47;
  then
A14: p in Int cell(GoB f,1,1+1) by A11,GOBOARD6:32;
A15: f/.(1+1) = (GoB f)*(1+1,1+1) by A10,MATRIX_0:50;
  1 < width GoB f by GOBOARD7:33;
  then
A16: 1+1 <= width GoB f by NAT_1:13;
  then
A17: [1+1,1+1] in Indices GoB f by A13,MATRIX_0:30;
  1 <= len GoB f by GOBOARD7:32;
  then
A18: [1,1+1] in Indices GoB f by A16,MATRIX_0:30;
A19: f/.1 = (GoB f)*(1,1+1) by A10,MATRIX_0:50;
  then right_cell(f,1) = cell(GoB f,1,1) by A8,A18,A17,A15,GOBOARD5:28;
  then
A20: Int cell(GoB f,1,1) c= RightComp f by GOBOARD9:def 2;
  left_cell(f,1) = cell(GoB f,1,1+1)by A8,A18,A19,A17,A15,GOBOARD5:28;
  then Int cell(GoB f,1,1+1) c= LeftComp f by GOBOARD9:def 1;
  then
A21: LC meets LeftComp f by A14,A12,XBOOLE_0:3;
A22: (f/.2)`1 = (E-max L~f)`1 by SPRECT_1:84
    .= E-bound L~f by EUCLID:52;
  set p = 1/2*((GoB f)*(1,1)+(GoB f)*(2,2));
A23: p in Int cell(GoB f,1,1) by A13,A11,GOBOARD6:31;
A24: p = 1/2*((GoB f)*(1,1)+f/.2) by A10,MATRIX_0:50
    .= 1/2*(f/.4+f/.2) by A10,MATRIX_0:50;
  then
A25: p`1 = 1/2*(f/.4+f/.2)`1 by TOPREAL3:4
    .= 1/2*((f/.4)`1+(f/.2)`1) by TOPREAL3:2
    .= 1/2*(f/.4)`1+ 1/2*(f/.2)`1;
  LeftComp f is_a_component_of (L~f)` by GOBOARD9:def 1;
  hence LeftComp f = LC by A4,A7,A21,GOBOARD9:1;
A26: 1/2*(W-bound L~f) + 1/2*(W-bound L~f) = (W-bound L~f);
A27: 1/2*(E-bound L~f) + 1/2*(E-bound L~f) = (E-bound L~f);
A28: 1/2*(N-bound L~f) + 1/2*(N-bound L~f) = (N-bound L~f);
A29: (f/.4)`1 = (W-min L~f)`1 by SPRECT_1:86
    .= W-bound L~f by EUCLID:52;
  then 1/2*(f/.4)`1 < 1/2*E-bound L~f by SPRECT_1:31,XREAL_1:68;
  then
A30: p`1 < E-bound L~f by A27,A25,A22,XREAL_1:6;
A31: p`2 = 1/2*(f/.4+f/.2)`2 by A24,TOPREAL3:4
    .= 1/2*((f/.4)`2+(f/.2)`2) by TOPREAL3:2
    .= 1/2*(f/.4)`2+ 1/2*(f/.2)`2;
  Lc = Lc9;
  then Rc is a_component by A5,A1,JORDAN1:36;
  then
A32: Rc9 is_a_component_of Lf` by CONNSP_1:def 6;
A33: (f/.2)`2 = (N-max L~f)`2 by SPRECT_1:84
    .= N-bound L~f by EUCLID:52;
A34: (f/.4)`2 = (S-min L~f)`2 by SPRECT_1:86
    .= S-bound L~f by EUCLID:52;
  then 1/2*(f/.4)`2 < 1/2*N-bound L~f by SPRECT_1:32,XREAL_1:68;
  then
A35: p`2 < N-bound L~f by A28,A31,A33,XREAL_1:6;
  1/2*(f/.2)`2 > 1/2*S-bound L~f by A33,SPRECT_1:32,XREAL_1:68;
  then
A36: S-bound L~f < p`2 by A6,A31,A34,XREAL_1:6;
  1/2*(f/.2)`1 > 1/2*W-bound L~f by A22,SPRECT_1:31,XREAL_1:68;
  then W-bound L~f < p`1 by A26,A25,A29,XREAL_1:6;
  then p in RC by A30,A36,A35;
  then
A37: RC meets RightComp f by A23,A20,XBOOLE_0:3;
  RightComp f is_a_component_of (L~f)` by GOBOARD9:def 2;
  hence thesis by A4,A32,A37,GOBOARD9:1;
end;
