
theorem Th7:
  for C be Simple_closed_curve for n,k be Nat st n
  is_sufficiently_large_for C holds 1 <= k & k+1 <= len Span(C,n) implies
right_cell(Span(C,n),k,Gauge(C,n)) misses C & left_cell(Span(C,n),k,Gauge(C,n))
  meets C
proof
  let C be Simple_closed_curve;
  let n,k be Nat;
  set G = Gauge(C,n);
  set f = Span(C,n);
  defpred P[Nat] means
for m be Nat st 1 <= m & m+1 <=
  len(f|$1) holds right_cell(f|$1,m,G) misses C & left_cell(f|$1,m,G) meets C;
A1: f|len f = f by FINSEQ_1:58;
  assume
A2: n is_sufficiently_large_for C;
  then
A3: f is_sequence_on G by JORDAN13:def 1;
A4: f/.2 = G*(X-SpanStart(C,n)-'1,Y-SpanStart(C,n)) by A2,JORDAN13:def 1;
A5: f/.1 = G*(X-SpanStart(C,n),Y-SpanStart(C,n)) by A2,JORDAN13:def 1;
A6: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A7: for m be Nat st 1 <= m & m+1 <= len(f|k) holds
    right_cell(f|k,m,G) misses C & left_cell(f|k,m,G) meets C;
A8: f|(k+1) is_sequence_on G by A3,GOBOARD1:22;
A9: f|k is_sequence_on G by A3,GOBOARD1:22;
    per cases;
    suppose
A10:  k >= len f;
      then
A11:  f|(k+1) = f by FINSEQ_1:58,NAT_1:12;
      f|k = f by A10,FINSEQ_1:58;
      hence thesis by A7,A11;
    end;
    suppose
A12:  k < len f;
      let m be Nat;
      assume that
A13:  1 <= m and
A14:  m+1 <= len(f|(k+1));
A15:  k+1 <= len f by A12,NAT_1:13;
      then
A16:  len(f|(k+1)) = k+1 by FINSEQ_1:59;
A17:  len(f|k) = k by A12,FINSEQ_1:59;
      now
        per cases by NAT_1:25;
        suppose
A18:      k = 0;
          1 <= m+1 by NAT_1:12;
          then m+1 = 0+1 by A14,A18,XXREAL_0:1;
          hence thesis by A13;
        end;
        suppose
A19:      k = 1;
          set j = Y-SpanStart(C,n);
          set i = X-SpanStart(C,n);
A20:      f|(k+1) = <*G*(i,j),G*(i-'1,j)*> by A5,A4,A15,A19,FINSEQ_5:81;
          then
A21:      (f|(k+1))/.1 = G*(i,j) by FINSEQ_4:17;
          1+1 <= m+1 by A13,XREAL_1:6;
          then
A22:      m+1 = 1+1 by A16,A14,A19,XXREAL_0:1;
A23:      [i-'1,j] in Indices G by A2,JORDAN11:9;
A24:      [i,j] in Indices G by A2,JORDAN11:8;
          i-'1 <= i by NAT_D:35;
          then
A25:      i+1 <> i-'1 by NAT_1:13;
A26:      j <> j+1;
A27:      (f|(k+1))/.2 = G*(i-'1,j) by A20,FINSEQ_4:17;
          then right_cell(f|(k+1),m,G) = cell(G,i-'1,j) by A8,A14,A21,A24,A23
,A22,A26,A25,GOBRD13:def 2;
          hence right_cell(f|(k+1),m,G) misses C by A2,JORDAN11:11;
          left_cell(f|(k+1),m,G) = cell(G,i-'1,j-'1) by A8,A14,A21,A27,A24,A23
,A22,A26,A25,GOBRD13:def 3;
          hence left_cell(f|(k+1),m,G) meets C by A2,JORDAN11:10;
        end;
        suppose
A28:      k > 1;
A29:      len(f|k) <= len f by FINSEQ_5:16;
A30:      1 <= (len(f|k))-'1 by A17,A28,NAT_D:49;
A31:      (len(f|k))-'1+1 = len(f|k) by A17,A28,XREAL_1:235;
          then
A32:      (len(f|k))-'1+(1+1) = (len(f|k))+1;
          now
            per cases;
            suppose
A33:          m+1 = len(f|(k+1));
              left_cell(f|k,(len(f|k))-'1,G) meets C by A7,A30,A31;
              then
A34:          left_cell(f,(len(f|k))-'1,G) meets C by A3,A17,A30,A31,A29,
GOBRD13:31;
              consider i1,j1,i2,j2 be Nat such that
A35:          [i1,j1] in Indices G and
A36:          f/.(len(f|k)-'1) = G*(i1,j1) and
A37:          [i2,j2] in Indices G and
A38:          f/.(len(f|k)) = G*(i2,j2) and
              i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1
              = j2 or i1 = i2 & j1 = j2+1 by A3,A12,A17,A30,A31,JORDAN8:3;
              1 <= i2 by A37,MATRIX_0:32;
              then
A39:          i2-'1+1 = i2 by XREAL_1:235;
A40:          1 <= j2 by A37,MATRIX_0:32;
              then
A41:          j2-'1+1 = j2 by XREAL_1:235;
              right_cell(f|k,(len(f|k))-'1,G) misses C by A7,A30,A31;
              then
A42:          right_cell(f,(len(f|k))-'1,G) misses C by A3,A17,A30,A31,A29,
GOBRD13:31;
              now
                per cases;
                suppose
A43:              front_right_cell(f,(len(f|k))-'1,G) misses C &
                  front_left_cell(f,(len(f|k))-'1,G) misses C;
                  then
A44:              f turns_left (len(f|k))-'1,G by A2,A15,A17,A30,A32,
JORDAN13:def 1;
                  now
                    per cases by A31,A35,A36,A37,A38,A44,GOBRD13:def 7;
                    suppose that
A45:                  i1 = i2 & j1+1 = j2 and
A46:                  [i2-'1,j2] in Indices G and
A47:                  f/.((len(f|k))-'1+2) = G*(i2-'1,j2);
                      cell(G,i1-'1,j1+1) misses C by A3,A30,A31,A29,A35,A36,A37
,A38,A43,A45,GOBRD13:34;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A39,A45,A46,A47,GOBRD13:26;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
A48:                  j1+1-'1 = j1 by NAT_D:34;
                      cell(G,i1-'1,j1) meets C by A3,A12,A17,A30,A31,A35,A36
,A37,A38,A34,A45,GOBRD13:21;
                      then
                      left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31,A33
,A37,A38,A39,A45,A46,A47,A48,GOBRD13:25;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                    suppose that
A49:                  i1+1 = i2 & j1 = j2 and
A50:                  [i2,j2+1] in Indices G and
A51:                  f/.((len(f|k))-'1+2) = G*(i2,j2+1);
                      cell(G,i2,j2) misses C by A3,A30,A31,A29,A35,A36,A37,A38
,A43,A49,GOBRD13:36;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A50,A51,GOBRD13:22;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
A52:                  i1+1-'1 = i1 by NAT_D:34;
                      cell(G,i1,j2) meets C by A3,A12,A17,A30,A31,A35,A36,A37
,A38,A34,A49,GOBRD13:23;
                      then
                      left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31,A33
,A37,A38,A49,A50,A51,A52,GOBRD13:21;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                    suppose that
A53:                  i1 = i2+1 & j1 = j2 and
A54:                  [i2,j2-'1] in Indices G and
A55:                  f/.((len(f|k))-'1+2) = G*(i2,j2-'1);
                      cell(G,i2-'1,j2-'1) misses C by A3,A30,A31,A29,A35,A36
,A37,A38,A43,A53,GOBRD13:38;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A41,A54,A55,GOBRD13:28;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i2,j2-'1) meets C by A3,A12,A17,A30,A31,A35,A36
,A37,A38,A34,A53,GOBRD13:25;
                      then
                      left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31,A33
,A37,A38,A41,A54,A55,GOBRD13:27;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                    suppose that
A56:                  i1 = i2 & j1 = j2+1 and
A57:                  [i2+1,j2] in Indices G and
A58:                  f/.((len(f|k))-'1+2) = G*(i2+1,j2);
                      cell(G,i1,j2-'1) misses C by A3,A30,A31,A29,A35,A36,A37
,A38,A43,A56,GOBRD13:40;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A56,A57,A58,GOBRD13:24;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i1,j2) meets C by A3,A12,A17,A30,A31,A35,A36,A37
,A38,A34,A56,GOBRD13:27;
                      then
                      left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31,A33
,A37,A38,A56,A57,A58,GOBRD13:23;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                  end;
                  hence thesis;
                end;
                suppose
A59:              front_right_cell(f,(len(f|k))-'1,G) misses C &
                  front_left_cell(f,(len(f|k))-'1,G) meets C;
                  then
A60:              f goes_straight (len(f|k))-'1,G by A2,A15,A17,A30,A32,
JORDAN13:def 1;
                  now
                    per cases by A31,A35,A36,A37,A38,A60,GOBRD13:def 8;
                    suppose that
A61:                  i1 = i2 & j1+1 = j2 and
A62:                  [i2,j2+1] in Indices G and
A63:                  f/.((len(f|k))-'1+2) = G*(i2,j2+1);
                      cell(G,i1,j1+1) misses C by A3,A30,A31,A29,A35,A36,A37
,A38,A59,A61,GOBRD13:35;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A61,A62,A63,GOBRD13:22;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i2-'1,j2) meets C by A3,A12,A17,A30,A31,A35,A36
,A37,A38,A59,A61,GOBRD13:34;
                      then left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A62,A63,GOBRD13:21;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                    suppose that
A64:                  i1+1 = i2 & j1 = j2 and
A65:                  [i2+1,j2] in Indices G and
A66:                  f/.((len(f|k))-'1+2) = G*(i2+1,j2);
                      cell(G,i1+1,j1-'1) misses C by A3,A30,A31,A29,A35,A36,A37
,A38,A59,A64,GOBRD13:37;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A64,A65,A66,GOBRD13:24;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i1+1,j1) meets C by A3,A12,A17,A30,A31,A35,A36,A37
,A38,A59,A64,GOBRD13:36;
                      then left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A64,A65,A66,GOBRD13:23;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                    suppose that
A67:                  i1 = i2+1 & j1 = j2 and
A68:                  [i2-'1,j2] in Indices G and
A69:                  f/.((len(f|k))-'1+2) = G*(i2-'1,j2);
                      cell(G,i2-'1,j2) misses C by A3,A30,A31,A29,A35,A36,A37
,A38,A59,A67,GOBRD13:39;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A39,A68,A69,GOBRD13:26;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i2-'1,j2-'1) meets C by A3,A12,A17,A30,A31,A35,A36
,A37,A38,A59,A67,GOBRD13:38;
                      then left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A39,A68,A69,GOBRD13:25;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                    suppose that
A70:                  i1 = i2 & j1 = j2+1 and
A71:                  [i2,j2-'1] in Indices G and
A72:                  f/.((len(f|k))-'1+2) = G*(i2,j2-'1);
A73:                  j2-'1+1=j2 by A40,XREAL_1:235;
                      cell(G,i2-'1,j2-'1) misses C by A3,A30,A31,A29,A35,A36
,A37,A38,A59,A70,GOBRD13:41;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A71,A72,A73,GOBRD13:28;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i2,j2-'1) meets C by A3,A12,A17,A30,A31,A35,A36
,A37,A38,A59,A70,GOBRD13:40;
                      then left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A41,A71,A72,GOBRD13:27;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                  end;
                  hence thesis;
                end;
                suppose
A74:              front_right_cell(f,(len(f|k))-'1,G) meets C;
                  then
A75:              f turns_right (len(f|k))-'1,G by A2,A15,A17,A30,A32,
JORDAN13:def 1;
                  now
                    per cases by A31,A35,A36,A37,A38,A75,GOBRD13:def 6;
                    suppose that
A76:                  i1 = i2 & j1+1 = j2 and
A77:                  [i2+1,j2] in Indices G and
A78:                  f/.((len(f|k))-'1+2) = G*(i2+1,j2);
A79:                  j1+1-'1 = j1 by NAT_D:34;
                      cell(G,i1,j1) misses C by A3,A30,A31,A29,A35,A36,A37,A38
,A42,A76,GOBRD13:22;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A76,A77,A78,A79,GOBRD13:24;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i2,j2) meets C by A3,A12,A17,A30,A31,A35,A36,A37
,A38,A74,A76,GOBRD13:35;
                      then left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A77,A78,GOBRD13:23;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                    suppose that
A80:                  i1+1 = i2 & j1 = j2 and
A81:                  [i2,j2-'1] in Indices G and
A82:                  f/.((len(f|k))-'1+2) = G*(i2,j2-'1);
A83:                  i1+1-'1 = i1 by NAT_D:34;
                      cell(G,i1,j1-'1) misses C by A3,A30,A31,A29,A35,A36,A37
,A38,A42,A80,GOBRD13:24;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A41,A80,A81,A82,A83,GOBRD13:28;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i2,j2-'1) meets C by A3,A12,A17,A30,A31,A35,A36
,A37,A38,A74,A80,GOBRD13:37;
                      then left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A41,A81,A82,GOBRD13:27;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                    suppose that
A84:                  i1 = i2+1 & j1 = j2 and
A85:                  [i2,j2+1] in Indices G and
A86:                  f/.((len(f|k))-'1+2) = G*(i2,j2+1);
                      cell(G,i2,j2) misses C by A3,A30,A31,A29,A35,A36,A37,A38
,A42,A84,GOBRD13:26;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A85,A86,GOBRD13:22;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i2-'1,j2) meets C by A3,A12,A17,A30,A31,A35,A36
,A37,A38,A74,A84,GOBRD13:39;
                      then left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A85,A86,GOBRD13:21;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                    suppose that
A87:                  i1 = i2 & j1 = j2+1 and
A88:                  [i2-'1,j2] in Indices G and
A89:                  f/.((len(f|k))-'1+2) = G*(i2-'1,j2);
                      cell(G,i2-'1,j2) misses C by A3,A30,A31,A29,A35,A36,A37
,A38,A42,A87,GOBRD13:28;
                      then right_cell(f,m,G) misses C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A39,A88,A89,GOBRD13:26;
                      hence
                      right_cell(f|(k+1),m,G) misses C by A3,A15,A16,A13,A33,
GOBRD13:31;
                      cell(G,i2-'1,j2-'1) meets C by A3,A12,A17,A30,A31,A35,A36
,A37,A38,A74,A87,GOBRD13:41;
                      then left_cell(f,m,G) meets C by A3,A15,A16,A17,A28,A31
,A33,A37,A38,A39,A88,A89,GOBRD13:25;
                      hence left_cell(f|(k+1),m,G) meets C by A3,A15,A16,A13
,A33,GOBRD13:31;
                    end;
                  end;
                  hence thesis;
                end;
              end;
              hence thesis;
            end;
            suppose
              m+1 <> len(f|(k+1));
              then m+1 < len(f|(k+1)) by A14,XXREAL_0:1;
              then
A90:          m+1 <= len(f|k) by A16,A17,NAT_1:13;
              then consider i1,j1,i2,j2 be Nat such that
A91:          [i1,j1] in Indices G and
A92:          (f|k)/.m = G*(i1,j1) and
A93:          [i2,j2] in Indices G and
A94:          (f|k)/.(m+1) = G*(i2,j2) and
A95:          i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+
              1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A9,A13,JORDAN8:3;
A96:          right_cell(f|k,m,G) misses C by A7,A13,A90;
A97:          f|(k+1) = (f|k)^<*f/.(k+1)*> by A15,FINSEQ_5:82;
              1 <= m+1 by NAT_1:12;
              then m+1 in dom(f|k) by A90,FINSEQ_3:25;
              then
A98:          (f|(k+1))/.(m+1) = G*(i2,j2) by A94,A97,FINSEQ_4:68;
A99:          left_cell(f|k,m,G) meets C by A7,A13,A90;
              m <= len(f|k) by A90,NAT_1:13;
              then m in dom(f|k) by A13,FINSEQ_3:25;
              then
A100:         (f|(k+1))/.m = G*(i1,j1) by A92,A97,FINSEQ_4:68;
              now
                per cases by A95;
                suppose
A101:             i1 = i2 & j1+1 = j2;
                  then
A102:             left_cell(f|k,m,G) = cell(G,i1-'1,j1) by A9,A13,A90,A91,A92
,A93,A94,GOBRD13:21;
                  right_cell(f|k,m,G) = cell(G,i1,j1) by A9,A13,A90,A91,A92,A93
,A94,A101,GOBRD13:22;
                  hence
                  thesis by A8,A13,A14,A91,A93,A96,A99,A100,A98,A101,A102,
GOBRD13:21,22;
                end;
                suppose
A103:             i1+1 = i2 & j1 = j2;
                  then
A104:             left_cell(f|k,m,G) = cell(G,i1,j1) by A9,A13,A90,A91,A92,A93
,A94,GOBRD13:23;
                  right_cell(f|k,m,G) = cell(G,i1,j1-'1) by A9,A13,A90,A91,A92
,A93,A94,A103,GOBRD13:24;
                  hence
                  thesis by A8,A13,A14,A91,A93,A96,A99,A100,A98,A103,A104,
GOBRD13:23,24;
                end;
                suppose
A105:             i1 = i2+1 & j1 = j2;
                  then
A106:             left_cell(f|k,m,G) = cell(G,i2,j2-'1) by A9,A13,A90,A91,A92
,A93,A94,GOBRD13:25;
                  right_cell(f|k,m,G) = cell(G,i2,j2) by A9,A13,A90,A91,A92,A93
,A94,A105,GOBRD13:26;
                  hence
                  thesis by A8,A13,A14,A91,A93,A96,A99,A100,A98,A105,A106,
GOBRD13:25,26;
                end;
                suppose
A107:             i1 = i2 & j1 = j2+1;
                  then
A108:             left_cell(f|k,m,G) = cell(G,i1,j2) by A9,A13,A90,A91,A92,A93
,A94,GOBRD13:27;
                  right_cell(f|k,m,G) = cell(G,i2-'1,j2) by A9,A13,A90,A91,A92
,A93,A94,A107,GOBRD13:28;
                  hence
                  thesis by A8,A13,A14,A91,A93,A96,A99,A100,A98,A107,A108,
GOBRD13:27,28;
                end;
              end;
              hence thesis;
            end;
          end;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
  end;
A109: P[0]
  by CARD_1:27;
  for k be Nat holds P[k] from NAT_1:sch 2(A109,A6);
  hence thesis by A1;
end;
