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 Th36:
  for f being rectangular special_circular_sequence holds GoB f =
  (f/.4,f/.1)][(f/.3,f/.2)
proof
  let f be rectangular special_circular_sequence;
  set G = (f/.4,f/.1)][(f/.3,f/.2), v1 = Incr X_axis f, v2 = Incr Y_axis f;
A1: f/.2 = N-max L~f by SPRECT_1:84;
A2: f/.1 = N-min L~f by SPRECT_1:83;
  then
A3: (f/.1)`1 < (f/.2)`1 by A1,SPRECT_2:51;
A4: (f/.2)`2 = (f/.1)`2 by A2,A1,PSCOMP_1:37;
A5: f/.4 = S-min L~f by SPRECT_1:86;
A6: f/.3 = S-max L~f by SPRECT_1:85;
  then
A7: (f/.3)`2 = (f/.4)`2 by A5,PSCOMP_1:53;
A8: len<*(f/.3)`1,(f/.4)`1,(f/.5)`1*> = 3 by FINSEQ_1:45;
A9: len f = 5 by SPRECT_1:82;
  then
A10: f/.1 = f/.5 by FINSEQ_6:def 1;
  set g = <*(f/.1)`1,(f/.2)`1*>;
  reconsider h = <*(f/.1)`1,(f/.2)`1*>^<*(f/.3)`1,(f/.4)`1 ,(f/.5)`1*>
     as FinSequence of REAL by RVSUM_1:145;
A11: f/.3 = E-min L~f by SPRECT_1:85;
A12: f/.2 = E-max L~f by SPRECT_1:84;
  then
A13: (f/.3)`1 = (f/.2)`1 by A11,PSCOMP_1:45;
A14: len g = 2 by FINSEQ_1:44;
A15: dom g = {1,2} by FINSEQ_1:2,89;
A16: g is increasing
  proof
    let n,m such that
A19: n in dom g and
A20: m in dom g and
A21: n<m;
A22: m = 1 or m = 2 by A15,A20,TARSKI:def 2;
    n = 1 or n = 2 by A15,A19,TARSKI:def 2;
    hence g.n < g.m by A2,A1,A21,A22,SPRECT_2:51;
  end;
A23: f/.4 = W-min L~f by SPRECT_1:86;
A24: len h = len <*(f/.1)`1,(f/.2)`1*> + len<*(f/.3)`1,(f/.4)`1,(f/.5)`1*>
  by FINSEQ_1:22
    .= len <*(f/.1)`1,(f/.2)`1*> + 3 by FINSEQ_1:45
    .= 2 + 3 by FINSEQ_1:44
    .= len f by SPRECT_1:82;
  for n st n in dom h holds h.n = (f/.n)`1
  proof
    let n;
    assume
A25: n in dom h;
    then
A26: 1 <= n by FINSEQ_3:25;
 n <= 5 by A9,A24,A25,FINSEQ_3:25;
    then n = 0 or ... or n = 5;
    then per cases by A26;
    suppose
A27:  n=1;
      then n in dom g by A14,FINSEQ_3:25;
      hence h.n = <*(f/.1)`1,(f/.2)`1*>.1 by A27,FINSEQ_1:def 7
        .= (f/.n)`1 by A27;
    end;
    suppose
A28:  n=2;
      then n in dom g by A14,FINSEQ_3:25;
      hence h.n = <*(f/.1)`1,(f/.2)`1*>.2 by A28,FINSEQ_1:def 7
        .= (f/.n)`1 by A28;
    end;
    suppose
A29:  n=3;
      hence h.n = h.(2+1)
        .= <*(f/.3)`1,(f/.4)`1,(f/.5)`1*>.1 by A14,A8,FINSEQ_1:65
        .= (f/.n)`1 by A29;
    end;
    suppose
A30:  n=4;
      hence h.n = h.(2+2)
        .= <*(f/.3)`1,(f/.4)`1,(f/.5)`1*>.2 by A14,A8,FINSEQ_1:65
        .= (f/.n)`1 by A30;
    end;
    suppose
A31:  n=5;
      hence h.n = h.(2+3)
        .= <*(f/.3)`1,(f/.4)`1,(f/.5)`1*>.3 by A14,A8,FINSEQ_1:65
        .= (f/.n)`1 by A31;
    end;
  end;
  then
A32: X_axis f = h by A24,GOBOARD1:def 1;
A33: rng g = { (f/.1)`1,(f/.2)`1 } by FINSEQ_2:127;
A34: f/.1 = W-max L~f by SPRECT_1:83;
  then
A35: (f/.4)`2 < (f/.5)`2 by A23,A10,SPRECT_2:57;
  reconsider vv1 = <*(f/.1)`1,(f/.2)`1*> as FinSequence of REAL by RVSUM_1:145;
  {(f/.3)`1,(f/.4)`1,(f/.5)`1} c= { (f/.1)`1,(f/.2)`1 }
  proof
    let x be object;
    assume
A36: x in {(f/.3)`1,(f/.4)`1,(f/.5)`1};
    per cases by A36,ENUMSET1:def 1;
    suppose
      x = (f/.3)`1;
      then x = (f/.2)`1 by A12,A11,PSCOMP_1:45;
      hence thesis by TARSKI:def 2;
    end;
    suppose
      x = (f/.4)`1;
      then x = (f/.1)`1 by A34,A23,PSCOMP_1:29;
      hence thesis by TARSKI:def 2;
    end;
    suppose
      x = (f/.5)`1;
      then x = (f/.1)`1 by A9,FINSEQ_6:def 1;
      hence thesis by TARSKI:def 2;
    end;
  end;
  then
A37: rng g = rng<*(f/.1)`1,(f/.2)`1*> \/ {(f/.3)`1,(f/.4)`1,(f/.5)`1} by A33,
XBOOLE_1:12
    .= rng<*(f/.1)`1,(f/.2)`1*> \/ rng<*(f/.3)`1,(f/.4)`1,(f/.5)`1*> by
FINSEQ_2:128
    .= rng X_axis f by A32,FINSEQ_1:31;
  len g = 2 by FINSEQ_1:44
    .= card rng X_axis f by A33,A37,A3,CARD_2:57;
  then
A38: v1 = vv1 by A37,A16,SEQ_4:def 21;
  then
A39: v1.1 = (f/.1)`1;
  set g = <*(f/.4)`2,(f/.5)`2*>;
  reconsider h = <*(f/.1)`2,(f/.2)`2,(f/.3)`2*>^<*(f/.4)`2,(f/.5)`2*>
     as FinSequence of REAL by RVSUM_1:145;
A40: len h = len <*(f/.1)`2,(f/.2)`2,(f/.3)`2*> + len<*(f/.4)`2,(f/.5)`2*>
  by FINSEQ_1:22
    .= len <*(f/.4)`2,(f/.5)`2*> + 3 by FINSEQ_1:45
    .= 2 + 3 by FINSEQ_1:44
    .= len f by SPRECT_1:82;
A41: len<*(f/.1)`2,(f/.2)`2,(f/.3)`2*> = 3 by FINSEQ_1:45;
A42: dom g = {1,2} by FINSEQ_1:2,89;
A43: g is increasing
  proof
    let n,m such that
A46: n in dom g and
A47: m in dom g and
A48: n<m;
A49: m = 1 or m = 2 by A42,A47,TARSKI:def 2;
    n = 1 or n = 2 by A42,A46,TARSKI:def 2;
    hence g.n < g.m by A34,A23,A10,A48,A49,SPRECT_2:57;
  end;
A50: v1.2 = (f/.2)`1 by A38;
A51: len g = 2 by FINSEQ_1:44;
  for n st n in dom h holds h.n = (f/.n)`2
  proof
    let n;
    assume
A52: n in dom h;
    then
A53: 1 <= n by FINSEQ_3:25;
 n <= 5 by A9,A40,A52,FINSEQ_3:25;
    then n = 0 or ... or n = 5;
    then per cases by A53;
    suppose
A54:  n=1;
      then n in dom<*(f/.1)`2,(f/.2)`2,(f/.3)`2*> by A41,FINSEQ_3:25;
      hence h.n = <*(f/.1)`2,(f/.2)`2,(f/.3)`2*>.1 by A54,FINSEQ_1:def 7
        .= (f/.n)`2 by A54;
    end;
    suppose
A55:  n=2;
      then n in dom<*(f/.1)`2,(f/.2)`2,(f/.3)`2*> by A41,FINSEQ_3:25;
      hence h.n = <*(f/.1)`2,(f/.2)`2,(f/.3)`2*>.2 by A55,FINSEQ_1:def 7
        .= (f/.n)`2 by A55;
    end;
    suppose
A56:  n=3;
      then n in dom<*(f/.1)`2,(f/.2)`2,(f/.3)`2*> by A41,FINSEQ_3:25;
      hence h.n = <*(f/.1)`2,(f/.2)`2,(f/.3)`2*>.3 by A56,FINSEQ_1:def 7
        .= (f/.n)`2 by A56;
    end;
    suppose
A57:  n=4;
      hence h.n = h.(3+1) .= <*(f/.4)`2,(f/.5)`2*>.1 by A51,A41,FINSEQ_1:65
        .= (f/.n)`2 by A57;
    end;
    suppose
A58:  n=5;
      hence h.n = h.(2+3) .= <*(f/.4)`2,(f/.5)`2*>.2 by A51,A41,FINSEQ_1:65
        .= (f/.n)`2 by A58;
    end;
  end;
  then
A59: Y_axis f = h by A40,GOBOARD1:def 2;
A60: rng g = { (f/.4)`2,(f/.5)`2 } by FINSEQ_2:127;
  {(f/.1)`2,(f/.2)`2,(f/.3)`2} c= { (f/.4)`2,(f/.5)`2 }
  proof
    let x be object;
    assume
A61: x in {(f/.1)`2,(f/.2)`2,(f/.3)`2};
    per cases by A61,ENUMSET1:def 1;
    suppose
      x = (f/.1)`2;
      hence thesis by A10,TARSKI:def 2;
    end;
    suppose
      x = (f/.2)`2;
      then x = (f/.1)`2 by A2,A1,PSCOMP_1:37;
      hence thesis by A10,TARSKI:def 2;
    end;
    suppose
      x = (f/.3)`2;
      then x = (f/.4)`2 by A6,A5,PSCOMP_1:53;
      hence thesis by TARSKI:def 2;
    end;
  end;
  then
A62: rng g = {(f/.1)`2,(f/.2)`2,(f/.3)`2} \/ {(f/.4)`2,(f/.5)`2} by A60,
XBOOLE_1:12
    .= rng<*(f/.1)`2,(f/.2)`2,(f/.3)`2*> \/ {(f/.4)`2,(f/.5)`2} by FINSEQ_2:128
    .= rng<*(f/.1)`2,(f/.2)`2,(f/.3)`2*> \/ rng<*(f/.4)`2,(f/.5)`2*> by
FINSEQ_2:127
    .= rng Y_axis f by A59,FINSEQ_1:31;
   reconsider vv2 = <*(f/.4)`2,(f/.1)`2*> as FinSequence of REAL
       by RVSUM_1:145;
  len g = 2 by FINSEQ_1:44
    .= card rng Y_axis f by A60,A62,A35,CARD_2:57;
  then
A63: v2 = vv2 by A10,A62,A43,SEQ_4:def 21;
  then
A64: v2.1 = (f/.4)`2;
A65: v2.2 = (f/.1)`2 by A63;
A66: (f/.1)`1 = (f/.4)`1 by A34,A23,PSCOMP_1:29;
A67: for n,m st [n,m] in Indices G holds G*(n,m) = |[v1.n,v2.m]|
  proof
    let n,m;
    assume [n,m] in Indices G;
    then
A68: [n,m]in { [1,1], [1,2],[2,1], [2,2]} by Th2;
    per cases by A68,ENUMSET1:def 2;
    suppose
A69:  [n,m] = [1,1];
      then
A70:  m = 1 by XTUPLE_0:1;
A71:  n = 1 by A69,XTUPLE_0:1;
      hence G*(n,m) = f/.4 by A70,MATRIX_0:50
        .= |[v1.n,v2.m]| by A39,A64,A66,A71,A70,EUCLID:53;
    end;
    suppose
A72:  [n,m] = [1,2];
      then
A73:  m = 2 by XTUPLE_0:1;
A74:  n = 1 by A72,XTUPLE_0:1;
      hence G*(n,m) = f/.1 by A73,MATRIX_0:50
        .= |[v1.n,v2.m]| by A39,A65,A74,A73,EUCLID:53;
    end;
    suppose
A75:  [n,m] = [2,1];
      then
A76:  m = 1 by XTUPLE_0:1;
A77:  n = 2 by A75,XTUPLE_0:1;
      hence G*(n,m) = f/.3 by A76,MATRIX_0:50
        .= |[v1.n,v2.m]| by A50,A64,A13,A7,A77,A76,EUCLID:53;
    end;
    suppose
A78:  [n,m] = [2,2];
      then
A79:  m = 2 by XTUPLE_0:1;
A80:  n = 2 by A78,XTUPLE_0:1;
      hence G*(n,m) = f/.2 by A79,MATRIX_0:50
        .= |[v1.n,v2.m]| by A50,A65,A4,A80,A79,EUCLID:53;
    end;
  end;
A81: width G = 2 by MATRIX_0:47
    .= len v2 by A63,FINSEQ_1:44;
  len G = 2 by MATRIX_0:47
    .= len v1 by A38,FINSEQ_1:44;
  then GoB(v1,v2) = G by A81,A67,GOBOARD2:def 1;
  hence thesis by GOBOARD2:def 2;
end;
