reserve f for non constant standard special_circular_sequence,
  i,j,k,i1,i2,j1,j2 for Nat,
  r,s,r1,s1,r2,s2 for Real,
  p,q for Point of TOP-REAL 2,
  G for Go-board;

theorem Th19:
  for k st 1 <= k & k+1 <= len f holds Int left_cell(f,k) c= LeftComp f
proof
  defpred P[Nat] means
  1 <= $1 & $1+1 <= len f implies Int left_cell(f,$1) c= LeftComp f;
A1: P[0];
A2: now
    let k such that
A3: P[k];
    thus P[k+1]
    proof
      assume that
A4:   1 <= k+1 and
A5:   k+1+1 <= len f;
      per cases by NAT_1:14;
      suppose k = 0;
        hence thesis by Def1;
      end;
      suppose
A6:     k >= 1;
A7:     k+1 <= len f by A5,NAT_1:13;
A8:     k <= k+1 by NAT_1:11;
        then k <= len f by A7,XXREAL_0:2;
        then
A9:     k in dom f by A6,FINSEQ_3:25;
        then consider i0,j0 being Nat such that
A10:    [i0,j0] in Indices GoB f and
A11:    f/.k = (GoB f)*(i0,j0) by GOBOARD2:14;
A12:    k+1 in dom f by A4,A7,FINSEQ_3:25;
        then consider i1,j1 being Nat such that
A13:    [i1,j1] in Indices GoB f and
A14:    f/.(k+1) = (GoB f)*(i1,j1) by GOBOARD2:14;
        k+1+1 >= 1 by NAT_1:11;
        then
A15:    k+1+1 in dom f by A5,FINSEQ_3:25;
        then consider i2,j2 being Nat such that
A16:    [i2,j2] in Indices GoB f and
A17:    f/.(k+1+1) = (GoB f)*(i2,j2) by GOBOARD2:14;
A18:    1 <= i0 by A10,MATRIX_0:32;
A19:    i0 <= len GoB f by A10,MATRIX_0:32;
A20:    1 <= i1 by A13,MATRIX_0:32;
A21:    i1 <= len GoB f by A13,MATRIX_0:32;
A22:    1 <= i2 by A16,MATRIX_0:32;
A23:    i2 <= len GoB f by A16,MATRIX_0:32;
A24:    1 <= j0 by A10,MATRIX_0:32;
A25:    j0 <= width GoB f by A10,MATRIX_0:32;
A26:    1 <= j1 by A13,MATRIX_0:32;
A27:    j1 <= width GoB f by A13,MATRIX_0:32;
A28:    1 <= j2 by A16,MATRIX_0:32;
A29:    j2 <= width GoB f by A16,MATRIX_0:32;
A30:    i0 = i1 or j0 = j1 by A9,A10,A11,A12,A13,A14,GOBOARD2:11;
A31:    i1 = i2 or j1 = j2 by A12,A13,A14,A15,A16,A17,GOBOARD2:11;
        reconsider i19=i1, i09=i0, i29=i2, j19=j1, j09=j0, j29=j2
        as Element of REAL by XREAL_0:def 1;
A32:    f is_sequence_on GoB f by GOBOARD5:def 5;
then |.i0-i1.|+|.j0-j1.| = 1 by A9,A10,A11,A12,A13,A14;
        then
A33:    |.i09-i19.|=1 & j0 = j1 or |.j09-j19.|=1 & i0=i1 by SEQM_3:42;
        then
A34:    i0 = i1 or i0 = i1+1 or i0+1 = i1 by SEQM_3:41;
|.i1-i2.|+|.j1-j2.| = 1 by A12,A13,A14,A15,A16,A17,A32;
        then
A35:    |.i19-i29.|=1 & j1 = j2 or |.j19-j29.|=1 & i1=i2 by SEQM_3:42;
        then
A36:    i1 = i2 or i1 = i2+1 or i1+1 = i2 by SEQM_3:41;
A37:    j0 = j1 or j0 = j1+1 or j0+1 = j1 by A33,SEQM_3:41;
A38:    j1 = j2 or j1 = j2+1 or j1+1 = j2 by A35,SEQM_3:41;
A39:    now
          assume that
A40:      i0 = i2 and
A41:      j0 = j2;
A42:      now
            assume k <= 1;
            then
A43:        k = 1 by A6,XXREAL_0:1;
            assume k+1+1 >= len f;
            then k+1+1 = len f by A5,XXREAL_0:1;
            hence contradiction by A43,GOBOARD7:34;
          end;
A44:      k < k+1+1 by A8,NAT_1:13;
          per cases by A42;
          suppose k > 1;
            hence contradiction by A5,A11,A17,A40,A41,A44,GOBOARD7:37;
          end;
          suppose k+1+1 < len f;
            hence contradiction by A6,A11,A17,A40,A41,A44,GOBOARD7:36;
          end;
        end;
        i1 >= 1 by A13,MATRIX_0:32;
        then
A45:    i1 = i1 -' 1 + 1 by XREAL_1:235;
        j1 >= 1 by A13,MATRIX_0:32;
        then
A46:    j1 = j1 -' 1 + 1 by XREAL_1:235;
        then j1 -' 1 <= j1 by NAT_1:11;
        then
A47:    j1 -' 1 <= width GoB f by A27,XXREAL_0:2;
        len GoB f > 1 by GOBOARD7:32;
        then
A48:    len GoB f -'1 + 1 = len GoB f by XREAL_1:235;
        width GoB f >= 1 by GOBOARD7:33;
        then
A49:    width GoB f -'1 + 1 = width GoB f by XREAL_1:235;
A50:    k+1+1 = k+(1+1);
A51:    i0+1+1 = i0+(1+1);
A52:    i2+1+1 = i2+(1+1);
A53:    j0+1+1 = j0+(1+1);
A54:    j2+1+1 = j2+(1+1);
A55:    LeftComp f is_a_component_of (L~f)` by Def1;
A56:    0+1 = 1;
now per cases by A9,A11,A12,A14,A15,A17,A34,A36,A37,A38,A39,GOBOARD7:29;
          suppose that
A57:        i0 = i1+1 and
A58:        i1 = i2+1 and
A59:        j0 = 1;
A60:        left_cell(f,k) = cell(GoB f,i1,0)
            by A6,A7,A10,A11,A13,A14,A30,A56,A57,A59,GOBOARD5:29;
            0+1 = j2 by A30,A31,A57,A58,A59;
            then
A61:        left_cell(f,k+1) = cell(GoB f,i2,0)
            by A4,A5,A13,A14,A16,A17,A30,A57,A58,A59,GOBOARD5:29;
A62:        LSeg(1/2*((GoB f)*(i1,1)+(GoB f)*(i0,1))- |[0,1]|,
            1/2*((GoB f)*(i2,1)+(GoB f)*(i1,1))- |[0,1]|)
            meets Int cell(GoB f,i1,0) by A19,A20,A57,GOBOARD6:84;
            LSeg(1/2*((GoB f)*(i1,1)+(GoB f)*(i0,1))- |[0,1]|,
            1/2*((GoB f)*(i2,1)+(GoB f)*(i1,1))- |[0,1]|) misses L~f
            by A19,A22,A52,A57,A58,GOBOARD8:25;
            then LSeg(1/2*((GoB f)*(i1,1)+(GoB f)*(i0,1))- |[0,1]|,
            1/2*((GoB f)*(i2,1)+(GoB f)*(i1,1))- |[0,1]|) c= (L~f)`
            by SUBSET_1:23;
            then
A63:        LSeg(1/2*((GoB f)*(i1,1)+(GoB f)*(i0,1))- |[0,1]|,
            1/2*((GoB f)*(i2,1)+(GoB f)*(i1,1))- |[0,1]|) c= LeftComp f
            by A3,A5,A6,A55,A60,A62,Th4,NAT_1:13;
A64:         Int left_cell(f,k+1) is convex by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A61,A63,A21,A22,A58,Th4,A64,GOBOARD6:84;
          end;
          suppose that
A65:        i0 = i1+1 and
A66:        i1 = i2+1 and
A67:        j0 <> 1;
A68:        left_cell(f,k) = cell(GoB f,i1,j1-'1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A65,GOBOARD5:29;
            1 < j0 by A24,A67,XXREAL_0:1;
            then
A69:        1 <= j0-'1 by A30,A46,A65,NAT_1:13;
A70:        left_cell(f,k+1) = cell(GoB f,i2,j1-'1)
            by A4,A5,A13,A14,A16,A17,A31,A46,A66,GOBOARD5:29;
A71:        LSeg(1/2*((GoB f)*(i1,j0-'1)+(GoB f)*(i0,j0)),
            1/2*((GoB f)*(i2,j0-'1)+(GoB f)*(i1,j0)))
            meets Int cell(GoB f,i1,j1-'1)
            by A19,A20,A27,A30,A46,A65,A69,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i1,j0-'1)+(GoB f)*(i0,j0)),
            1/2*((GoB f)*(i2,j0-'1)+(GoB f)*(i1,j0))) misses L~f
            by A5,A6,A11,A14,A17,A19,A22,A25,A30,A31,A46,A50,A52,A65,A66,A69,
GOBOARD8:11;
            then LSeg(1/2*((GoB f)*(i1,j0-'1)+(GoB f)*(i0,j0)),
            1/2*((GoB f)*(i2,j0-'1)+(GoB f)*(i1,j0))) c= (L~f)` by SUBSET_1:23;
            then
A72:        LSeg(1/2*((GoB f)*(i1,j0-'1)+(GoB f)*(i0,j0)),
            1/2*((GoB f)*(i2,j0-'1)+(GoB f)*(i1,j0))) c= LeftComp f
            by A3,A5,A6,A55,A68,A71,Th4,NAT_1:13;
A73:         Int left_cell(f,k+1) is convex
                by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A30,A55,A65,A66,A70,A72,A21,A22,A25,A46,A69,Th4,A73
,GOBOARD6:82;
          end;
          suppose that
A74:        i0 = i1+1 and
A75:        j1 = j2+1;
            left_cell(f,k) = cell(GoB f,i1,j2)
            by A6,A7,A10,A11,A13,A14,A30,A74,A75,GOBOARD5:29
.= left_cell(f,k+1) by A4,A5,A13,A14,A16,A17,A31,A45,A75,GOBOARD5:30;
            hence thesis by A3,A5,A6,NAT_1:13;
          end;
          suppose that
A76:        j0 = j1+1 and
A77:        i1 = i2+1 and
A78:        i0 = len GoB f and
A79:        j1 = 1;
A80:        left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A45,A76,GOBOARD5:30;
A81:        LSeg((GoB f)*(len GoB f,1)+|[1,-1]|,
            1/2*((GoB f)*(len GoB f,1)+(GoB f)*(len GoB f,2))+|[1,0]|)
            meets Int cell(GoB f,i0,j1) by A25,A76,A78,A79,GOBOARD6:86;
            LSeg((GoB f)*(len GoB f,1)+|[1,-1]|,
            1/2*((GoB f)*(len GoB f,1)+(GoB f)*(len GoB f,2))+|[1,0]|)
            misses L~f by GOBOARD8:35;
            then LSeg((GoB f)*(len GoB f,1)+|[1,-1]|,
            1/2*((GoB f)*(len GoB f,1)+(GoB f)*(len GoB f,2))+|[1,0]|)
            c= (L~f)` by SUBSET_1:23;
            then
A82:        LSeg((GoB f)*(len GoB f,1)+|[1,-1]|,
            1/2*((GoB f)*(len GoB f,1)+(GoB f)*(len GoB f,2))+|[1,0]|)
            c= LeftComp f by A3,A5,A6,A55,A80,A81,Th4,NAT_1:13;
A83:         Int cell(GoB f,i1,0) is convex by A21,A26,A27,Th16;
            Int cell(GoB f,i1,0) misses L~f by A21,A26,A27,GOBOARD7:12;
            then Int cell(GoB f,i1,0) c= (L~f)` by SUBSET_1:23;
            then
A84:        Int cell(GoB f,i1,0) c= LeftComp f
              by A55,A82,A30,A76,A78,Th4,A83,GOBOARD6:90;
A85:        left_cell(f,k+1) = cell(GoB f,i2,0)
            by A4,A5,A13,A14,A16,A17,A31,A46,A77,A79,GOBOARD5:29;
A86:        LSeg
            (1/2*((GoB f)*(len GoB f -' 1,1)+(GoB f)*(len GoB f,1))- |[0,1]|,
            (GoB f)*(len GoB f,1)+|[1,-1]|)
            meets Int cell(GoB f,i1,0) by A30,A76,A78,GOBOARD6:90;
LSeg(1/2*((GoB f)*(len GoB f -' 1,1)+(GoB f)*(len GoB f,1))- |[0,1]|,
            (GoB f)*(len GoB f,1)+|[1,-1]|) misses L~f by GOBOARD8:27;
            then LSeg(1/2*((GoB f)*(len GoB f -' 1,1)+(GoB f)*
            (len GoB f,1))- |[0,1]|,
            (GoB f)*(len GoB f,1)+|[1,-1]|) c= (L~f)` by SUBSET_1:23;
            then
A87:        LSeg(1/2*((GoB f)*(len GoB f -' 1,1)+(GoB f)*
              (len GoB f,1))- |[0,1]|,
              (GoB f)*(len GoB f,1)+|[1,-1]|) c= LeftComp f
            by A55,A84,A86,Th4;
A88:         Int left_cell(f,k+1) is convex by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A85,A87,A22,A30,A45,A76,A77,A78,Th4,A88,
GOBOARD6:84;
          end;
          suppose that
A89:        j0 = j1+1 and
A90:        i1 = i2+1 and
A91:        i0 <> len GoB f and
A92:        j1 = 1;
A93:        left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A45,A89,GOBOARD5:30;
            len GoB f > i0 by A19,A91,XXREAL_0:1;
            then
A94:        len GoB f >= i0+1 by NAT_1:13;
A95:        LSeg(1/2*((GoB f)*(i0,1)+(GoB f)*(i0+1,1))- |[0,1]|,
            1/2*((GoB f)*(i1,1)+(GoB f)*(i1+1,1+1)))
            meets Int cell(GoB f,i0,j1)
            by A20,A25,A30,A89,A92,A94,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i0,1)+(GoB f)*(i0+1,1))- |[0,1]|,
            1/2*((GoB f)*(i1,1)+(GoB f)*(i1+1,2))) misses L~f
by A5,A6,A11,A14,A17,A22,A30,A31,A50,A52,A89,A90,A92,A94,GOBOARD8:8;
            then LSeg(1/2*((GoB f)*(i0,1)+(GoB f)*(i0+1,1))- |[0,1]|,
            1/2*((GoB f)*(i1,1)+(GoB f)*(i1+1,2))) c= (L~f)` by SUBSET_1:23;
            then
A96:       LSeg(1/2*((GoB f)*(i0,1)+(GoB f)*(i0+1,1))- |[0,1 ]|,
            1/2*((GoB f)*(i1,1)+(GoB f)*(i1+1,2))) c= LeftComp f
            by A3,A5,A6,A55,A93,A95,Th4,NAT_1:13;
A97:         Int cell(GoB f,i1,0) is convex
             by A21,A26,A27,Th16;
            Int cell(GoB f,i1,0) misses L~f by A21,A26,A27,GOBOARD7:12;
            then Int cell(GoB f,i1,0) c= (L~f)` by SUBSET_1:23;
            then
A98:       Int cell(GoB f,i1,0) c= LeftComp f by A55,A96,A20,A30,A89,A94,Th4
,A97,GOBOARD6:84;
A99:       left_cell(f,k+1) = cell(GoB f,i2,0)
            by A4,A5,A13,A14,A16,A17,A31,A46,A90,A92,GOBOARD5:29;
A100:       LSeg(1/2*((GoB f)*(i2,1)+(GoB f)*(i2+1,1))- |[0,1]|,
            1/2*((GoB f)*(i2+1,1)+(GoB f)*(i2+2,1))- |[0,1]|)
            meets Int cell(GoB f,i1,0) by A20,A30,A89,A90,A94,GOBOARD6:84;
            LSeg(1/2*((GoB f)*(i2,1)+(GoB f)*(i2+1,1))- |[0,1]|,
            1/2*((GoB f)*(i2+1,1)+(GoB f)*(i2+2,1))- |[0,1]|) misses L~f
            by A22,A30,A89,A90,A94,GOBOARD8:25;
            then LSeg(1/2*((GoB f)*(i2,1)+(GoB f)*(i2+1,1))- |[0,1]|,
            1/2*((GoB f)*(i2+1,1)+(GoB f)*(i2+2,1))- |[0,1]|) c= (L~f)`
            by SUBSET_1:23;
            then
A101:       LSeg(1/2*((GoB f)*(i2,1)+(GoB f)*(i2+1,1))- |[0,1 ]|,
            1/2*((GoB f)*(i2+1,1)+(GoB f)*(i2+2,1))- |[0,1]|) c= LeftComp f
            by A55,A98,A100,Th4;
A102:         Int left_cell(f,k+1) is convex
              by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A99,A101,A21,A22,A90,Th4,A102,GOBOARD6:84;
          end;
          suppose that
A103:       j0 = j1+1 and
A104:       i1 = i2+1 and
A105:       i0 = len GoB f and
A106:       j1 <> 1;
A107:       left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A45,A103,GOBOARD5:30;
            1 < j1 by A26,A106,XXREAL_0:1;
            then
A108:       1 <= j1-'1 by A46,NAT_1:13;
A109:       j1+1 = j1-'1+(1+1) by A46;
A110:       LSeg
            (1/2*((GoB f)*(len GoB f,j1-'1)+(GoB f)*(len GoB f,j1))+|[1,0]|,
            1/2*((GoB f)*(len GoB f,j1)+(GoB f)*(len GoB f,j1+1))+|[1,0]|)
            meets Int cell(GoB f,i0,j1) by A25,A26,A103,A105,GOBOARD6:86;
LSeg(1/2*((GoB f)*(len GoB f,j1-'1)+(GoB f)*(len GoB f,j1))+|[1,0]|,
            1/2*((GoB f)*(len GoB f,j1)+(GoB f)*(len GoB f,j1+1))+|[1,0]|)
            misses L~f by A25,A46,A103,A108,A109,GOBOARD8:34;
            then LSeg(1/2*((GoB f)*(len GoB f,j1-'1)+(GoB f)*
            (len GoB f,j1))+|[1,0]|,
            1/2*((GoB f)*(len GoB f,j1)+(GoB f)*(len GoB f,j1+1))+|[1,0]|)
            c= (L~f)` by SUBSET_1:23;
            then
            A111:       LSeg
(1/2*((GoB f)*(len GoB f,j1-'1)+(GoB f)*(len GoB f,j1))+|[1
            ,0]|,
            1/2*((GoB f)*(len GoB f,j1)+(GoB f)*(len GoB f,j1+1))+|[1,0]|)
            c= LeftComp f by A3,A5,A6,A55,A107,A110,Th4,NAT_1:13;
            j1 > 1 by A26,A106,XXREAL_0:1;
            then
A112:           1 <= j1-'1 by A46,NAT_1:13;
A113:         Int cell(GoB f,i1,j1-'1) is convex by A21,A47,Th16;
            Int cell(GoB f,i1,j1-'1) misses L~f by A21,A47,GOBOARD7:12;
            then Int cell(GoB f,i1,j1-'1) c= (L~f)` by SUBSET_1:23;
            then
A114:       Int cell(GoB f,i1,j1-'1) c= LeftComp f by A55,A111,A27,A30,A46,A103
,A105,Th4,A113,A112,GOBOARD6:86;
A115:       left_cell(f,k+1) = cell(GoB f,i2,j1-'1)
            by A4,A5,A13,A14,A16,A17,A31,A46,A104,GOBOARD5:29;
A116:       LSeg(1/2*((GoB f)*(i1 -' 1,j1-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1,j1-'1)+(GoB f)*(i1,j1))+|[1,0]|)
            meets Int cell(GoB f,i1,j1-'1) by A27,A30,A46,A103,A105,A108,
GOBOARD6:86;
A117:       i1-'1 = i2 by A104,NAT_D:34;
            then LSeg(1/2*((GoB f)*(i1 -' 1,j1-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1,j1-'1)+(GoB f)*(i1,j1))+|[1,0]|) misses L~f
            by A5,A6,A11,A14,A17,A25,A30,A31,A46,A50,A103,A104,A105,A108,A109,
GOBOARD8:19;
            then LSeg(1/2*((GoB f)*(i1 -' 1,j1-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1,j1-'1)+(GoB f)*(i1,j1))+|[1,0]|) c= (L~f)`
            by SUBSET_1:23;
            then
A118:       LSeg(1/2*((GoB f)*(i1 -' 1,j1-'1)+(GoB f)*(i1,j1)) ,
            1/2*((GoB f)*(i1,j1-'1)+(GoB f)*(i1,j1))+|[1,0]|) c= LeftComp f
            by A55,A114,A116,Th4;
A119:         Int left_cell(f,k+1) is convex
                  by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A115,A118,A21,A22,A27,A46,A104,A108,A117,Th4
,A119,GOBOARD6:82;
          end;
          suppose that
A120:       j0 = j1+1 and
A121:       i1 = i2+1 and
A122:       i0 <> len GoB f and
A123:       j1 <> 1;
A124:       left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A45,A120,GOBOARD5:30;
            1 < j1 by A26,A123,XXREAL_0:1;
            then
A125:       1 <= j1-'1 by A46,NAT_1:13;
            len GoB f > i0 by A19,A122,XXREAL_0:1;
            then
A126:       len GoB f >= i0+1 by NAT_1:13;
A127:       j1+1 = j1-'1+(1+1) by A46;
A128:       LSeg(1/2*((GoB f)*(i0,j1-'1)+(GoB f)*(i0+1,j1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1+1,j1+1)))
meets Int cell(GoB f,i0,j1) by A20,A25,A26,A30,A120,A126,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i0,j1-'1)+(GoB f)*(i0+1,j1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1+1,j1+1))) misses L~f
by A5,A6,A11,A14,A17,A22,A25,A30,A31,A46,A50,A52,A120,A121,A125,A126,A127,
GOBOARD8:5;
            then LSeg(1/2*((GoB f)*(i0,j1-'1)+(GoB f)*(i0+1,j1)),
1/2*((GoB f)*(i1,j1)+(GoB f)*(i1+1,j1+1))) c= (L~f)` by SUBSET_1:23;
            then
A129:       LSeg(1/2*((GoB f)*(i0,j1-'1)+(GoB f)*(i0+1,j1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1+1,j1+1))) c= LeftComp f
            by A3,A5,A6,A55,A124,A128,Th4,NAT_1:13;
            j1 > 1 by A26,A123,XXREAL_0:1;
            then
A130:            1 <= j1-'1 by A46,NAT_1:13;
A131:         Int cell(GoB f,i1,j1-'1) is convex by A21,A47,Th16;
            Int cell(GoB f,i1,j1-'1) misses L~f by A21,A47,GOBOARD7:12;
            then Int cell(GoB f,i1,j1-'1) c= (L~f)` by SUBSET_1:23;
            then
A132:       Int cell(GoB f,i1,j1-'1) c= LeftComp f by A30,A55,A120,A129,A20,A27
,A46,A126,Th4,A131,A130,GOBOARD6:82;
A133:       left_cell(f,k+1) = cell(GoB f,i2,j1-'1)
            by A4,A5,A13,A14,A16,A17,A31,A46,A121,GOBOARD5:29;
A134:       LSeg(1/2*((GoB f)*(i1,j1-'1)+(GoB f)*(i1+1,j1)),
            1/2*((GoB f)*(i2,j1-'1)+(GoB f)*(i1,j1)))
            meets Int cell(GoB f,i1,j1-'1) by A20,A27,A30,A46,A120,A125,A126,
GOBOARD6:82;
            i1 < len GoB f by A21,A30,A120,A122,XXREAL_0:1;
            then i1+1 <= len GoB f by NAT_1:13;
            then LSeg(1/2*((GoB f)*(i1,j1-'1)+(GoB f)*(i1+1,j1)),
            1/2*((GoB f)*(i2,j1-'1)+(GoB f)*(i1,j1))) misses L~f
by A5,A6,A11,A14,A17,A22,A25,A30,A31,A46,A50,A52,A120,A121,A125,A127,
GOBOARD8:13;
            then LSeg(1/2*((GoB f)*(i1,j1-'1)+(GoB f)*(i1+1,j1)),
            1/2*((GoB f)*(i2,j1-'1)+(GoB f)*(i1,j1))) c= (L~f)` by SUBSET_1:23;
            then
A135:       LSeg(1/2*((GoB f)*(i1,j1-'1)+(GoB f)*(i1+1,j1)),
            1/2*((GoB f)*(i2,j1-'1)+(GoB f)*(i1,j1))) c= LeftComp f
            by A55,A132,A134,Th4;
A136:         Int left_cell(f,k+1) is convex
                by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A121,A133,A135,A21,A22,A27,A46,A125,Th4,A136,
GOBOARD6:82;
          end;
          suppose that
A137:       j0 = j1+1 and
A138:       j1 = j2+1 and
A139:       i0 = len GoB f;
A140:       left_cell(f,k) = cell(GoB f,len GoB f,j1)
            by A6,A7,A10,A11,A13,A14,A30,A48,A137,A139,GOBOARD5:30;
A141:       left_cell(f,k+1) = cell(GoB f,len GoB f,j2)
            by A4,A5,A13,A14,A16,A17,A30,A31,A48,A137,A138,A139,GOBOARD5:30;
A142:       LSeg(1/2*((GoB f)*(len GoB f,j2)+(GoB f)*(len GoB f,j2+1))+|[1,0]|,
            1/2*((GoB f)*(len GoB f,j2+1)+(GoB f)*(len GoB f,j2+2))+|[1,0]|)
            meets Int cell(GoB f,len GoB f,j1)
            by A25,A26,A137,A138,GOBOARD6:86;
            LSeg(1/2*((GoB f)*(len GoB f,j2)+(GoB f)*(len GoB f,j2+1))+|[1,0]|
            , 1/2*((GoB f)*(len GoB f,j2+1)+(GoB f)*(len GoB f,j2+2))+|[1,0]|)
            misses L~f by A25,A28,A137,A138,GOBOARD8:34;
then LSeg(1/2*((GoB f)*(len GoB f,j2)+(GoB f)*(len GoB f,j2+1))+|[1,0]|,
            1/2*((GoB f)*(len GoB f,j2+1)+(GoB f)* (len GoB f,j2+2))+|[1,0]|)
            c= (L~f)` by SUBSET_1:23;
            then
            A143:       LSeg
(1/2*((GoB f)*(len GoB f,j2)+(GoB f)*(len GoB f,j2+1))+|[1,
            0]|,
            1/2*((GoB f)*(len GoB f,j2+1)+(GoB f)*(len GoB f,j2+2))+|[1,0]|)
            c= LeftComp f by A3,A5,A6,A55,A140,A142,Th4,NAT_1:13;
A144:         Int left_cell(f,k+1) is convex
              by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A141,A143,A27,A28,A138,Th4,A144,GOBOARD6:86;
          end;
          suppose that
A145:       j0 = j1+1 and
A146:       j1 = j2+1 and
A147:       i0 <> len GoB f;
A148:       left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A45,A145,GOBOARD5:30;
            len GoB f > i0 by A19,A147,XXREAL_0:1;
            then
A149:       len GoB f >= i0+1 by NAT_1:13;
A150:       left_cell(f,k+1) = cell(GoB f,i1,j2)
            by A4,A5,A13,A14,A16,A17,A31,A45,A146,GOBOARD5:30;
A151:       LSeg(1/2*((GoB f)*(i0,j2)+(GoB f)*(i0+1,j2+1)),
            1/2*((GoB f)*(i1,j2+1)+(GoB f)*(i1+1,j2+2)))
            meets Int cell(GoB f,i0,j1)
            by A20,A25,A26,A30,A145,A146,A149,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i0,j2)+(GoB f)*(i0+1,j2+1)),
            1/2*((GoB f)*(i1,j2+1)+(GoB f)*(i1+1,j2+2))) misses L~f
by A5,A6,A11,A14,A17,A18,A25,A28,A30,A31,A50,A145,A146,A149,GOBOARD8:4;
            then LSeg(1/2*((GoB f)*(i0,j2)+(GoB f)*(i0+1,j2+1)),
            1/2*((GoB f)*(i1,j2+1)+(GoB f)*
            (i1+1,j2+2))) c= (L~f)` by SUBSET_1:23;
            then
A152:       LSeg(1/2*((GoB f)*(i0,j2)+(GoB f)*(i0+1,j2+1)),
            1/2*((GoB f)*(i1,j2+1)+(GoB f)*(i1+1,j2+2))) c= LeftComp f
            by A3,A5,A6,A55,A148,A151,Th4,NAT_1:13;
A153:         Int left_cell(f,k+1) is convex
               by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A150,A152,A20,A27,A28,A30,A145,A146,A149,Th4
,A153,GOBOARD6:82;
          end;
          suppose that
A154:       i0+1 = i1 and
A155:       j1 = j2+1 and
A156:       i1 = len GoB f and
A157:       j0 = width GoB f;
A158:       left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A154,GOBOARD5:28;
A159:       LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i1,j0))+|[0,1]|,
            (GoB f)*(i1,j0)+|[1,1]|)
            meets Int cell(GoB f,i0,j1) by A18,A21,A30,A154,A157,GOBOARD6:83;
            LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i1,j0))+|[0,1]|,
            (GoB f)*(i1,j0)+|[1,1]|)
            misses L~f by A48,A154,A156,A157,GOBOARD8:30;
            then LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i1,j0))+|[0,1]|,
            (GoB f)*(i1,j0)+|[1,1]|) c= (L~f)` by SUBSET_1:23;
            then
A160:       LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i1,j0))+|[0,1]| ,
(GoB f)*(i1,j0)+|[1,1]|) c= LeftComp f by A3,A5,A6,A55,A158,A159,Th4,NAT_1:13;
A161:         Int cell(GoB f,i1,j1) is convex by A21,A27,Th16;
            Int cell(GoB f,i1,j1) misses L~f by A21,A27,GOBOARD7:12;
            then Int cell(GoB f,i1,j1) c= (L~f)` by SUBSET_1:23;
            then
A162:       Int cell(GoB f,i1,j1) c= LeftComp f by A55,A160,A30,A154,A156,A157
,Th4,A161,GOBOARD6:88;
A163:       left_cell(f,k+1) = cell(GoB f,i1,j2)
            by A4,A5,A13,A14,A16,A17,A31,A45,A155,GOBOARD5:30;
A164:       LSeg(1/2*((GoB f)*(i1,j2)+(GoB f)*(i1,j1))+|[1,0]|,
            (GoB f)*(i1,j1)+|[1,1]|) meets Int cell(GoB f,i1,j1)
            by A30,A154,A156,A157,GOBOARD6:88;
            LSeg(1/2*((GoB f)*(i1,j2)+(GoB f)*(i1,j1))+|[1,0]|, (GoB f)*
(i1,j1)+|[1,1]|) misses L~f by A30,A49,A154,A155,A156,A157,GOBOARD8:36;
            then LSeg(1/2*((GoB f)*(i1,j2)+(GoB f)*(i1,j1))+|[1,0]|,
            (GoB f)*(i1,j1)+|[1,1]|) c= (L~f)` by SUBSET_1:23;
            then
A165:       LSeg(1/2*((GoB f)*(i1,j2)+(GoB f)*(i1,j1))+|[1,0]| ,
            (GoB f)*(i1,j1)+|[1,1]|) c= LeftComp f by A55,A162,A164,Th4;
A166:         Int left_cell(f,k+1) is convex
                by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A163,A165,A27,A28,A155,A156,Th4,A166,
GOBOARD6:86;
          end;
          suppose that
A167:       i0+1 = i1 and
A168:       j1 = j2+1 and
A169:       i1 <> len GoB f and
A170:       j0 = width GoB f;
A171:       left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A167,GOBOARD5:28;
            len GoB f > i1 by A21,A169,XXREAL_0:1;
            then
A172:       i1+1 <= len GoB f by NAT_1:13;
A173:       LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0+1,j0))+|[0,1]|,
            1/2*((GoB f)*(i0+1,j0)+(GoB f)*(i0+2,j0))+|[0,1]|)
            meets Int cell(GoB f,i0,j1) by A18,A21,A30,A167,A170,GOBOARD6:83;
            LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0+1,j0))+|[0,1]|,
            1/2*((GoB f)*(i0+1,j0)+(GoB f)*(i0+2,j0))+|[0,1]|)
            misses L~f by A18,A167,A170,A172,GOBOARD8:28;
            then LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0+1,j0))+|[0,1]|,
            1/2*((GoB f)*(i0+1,j0)+(GoB f)*(i0+2,j0))+|[0,1]|)
            c= (L~f)` by SUBSET_1:23;
            then
A174:       LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0+1,j0))+|[0,1 ]|,
            1/2*((GoB f)*(i0+1,j0)+(GoB f)*(i0+2,j0))+|[0,1]|) c= LeftComp f
            by A3,A5,A6,A55,A171,A173,Th4,NAT_1:13;
A175:         Int cell(GoB f,i1,j1) is convex by A21,A27,Th16;
            Int cell(GoB f,i1,j1) misses L~f by A21,A27,GOBOARD7:12;
            then Int cell(GoB f,i1,j1) c= (L~f)` by SUBSET_1:23;
            then
A176:       Int cell(GoB f,i1,j1) c= LeftComp f by A55,A174,A20,A30,A167,A170
,A172,Th4,A175,GOBOARD6:83;
A177:       left_cell(f,k+1) = cell(GoB f,i1,j2)
            by A4,A5,A13,A14,A16,A17,A31,A45,A168,GOBOARD5:30;
A178:       LSeg(1/2*((GoB f)*(i0+1,j2)+(GoB f)*(i1+1,j0)),
            1/2*((GoB f)*(i0+1,j0)+(GoB f)*(i1+1,j0))+|[0,1]|)
            meets Int cell(GoB f,i1,j1) by A20,A30,A167,A170,A172,GOBOARD6:83;
            LSeg(1/2*((GoB f)*(i0+1,j2)+(GoB f)*(i1+1,j0)),
            1/2*((GoB f)*(i0+1,j0)+(GoB f)*(i1+1,j0))+|[0,1]|) misses L~f
            by A5,A6,A11,A14,A17,A18,A30,A31,A49,A50,A51,A167,A168,A170,A172,
GOBOARD8:10;
            then LSeg(1/2*((GoB f)*(i0+1,j2)+(GoB f)*(i1+1,j0)),
            1/2*((GoB f)*(i0+1,j0)+(GoB f)*(i1+1,j0))+|[0,1]|) c= (L~f)`
            by SUBSET_1:23;
            then
A179:       LSeg(1/2*((GoB f)*(i0+1,j2)+(GoB f)*(i1+1,j0)),
            1/2*((GoB f)*(i0+1,j0)+(GoB f)*(i1+1,j0))+|[0,1]|) c= LeftComp f
            by A55,A176,A178,Th4;
A180:         Int left_cell(f,k+1) is convex
                by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A177,A179,A20,A28,A30,A167,A168,A170,A172,Th4
,A180,GOBOARD6:82;
          end;
          suppose that
A181:       i0+1 = i1 and
A182:       j1 = j2+1 and
A183:       i1 = len GoB f and
A184:       j0 <> width GoB f;
A185:       left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A181,GOBOARD5:28;
            width GoB f > j0 by A25,A184,XXREAL_0:1;
            then
A186:       width GoB f >= j0+1 by NAT_1:13;
A187:       LSeg(1/2*((GoB f)*(i0,j1)+(GoB f)*(i1,j1+1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1,j1+1))+|[1,0]|)
meets Int cell(GoB f,i0,j1) by A18,A21,A26,A30,A181,A186,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i0,j1)+(GoB f)*(i1,j1+1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1,j1+1))+|[1,0]|) misses L~f
            by A5,A6,A11,A14,A17,A28,A30,A31,A48,A50,A54,A181,A182,A183,A186,
GOBOARD8:20;
            then LSeg(1/2*((GoB f)*(i0,j1)+(GoB f)*(i1,j1+1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1,j1+1))+|[1,0]|)
            c= (L~f)` by SUBSET_1:23;
            then
A188:       LSeg(1/2*((GoB f)*(i0,j1)+(GoB f)*(i1,j1+1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1,j1+1))+|[1,0]|) c= LeftComp f
            by A3,A5,A6,A55,A185,A187,Th4,NAT_1:13;
A189:         Int cell(GoB f,i1,j1) is convex by A21,A27,Th16;
            Int cell(GoB f,i1,j1) misses L~f by A21,A27,GOBOARD7:12;
            then Int cell(GoB f,i1,j1) c= (L~f)` by SUBSET_1:23;
            then
A190:       Int cell(GoB f,i1,j1) c= LeftComp f by A55,A188,A26,A30,A181,A183
,A186,Th4,A189,GOBOARD6:86;
A191:       left_cell(f,k+1) = cell(GoB f,i1,j2)
            by A4,A5,A13,A14,A16,A17,A31,A45,A182,GOBOARD5:30;
A192:       LSeg(1/2*((GoB f)*(len GoB f,j2)+(GoB f)*(len GoB f,j1))+|[1,0]|,
            1/2*((GoB f)*(len GoB f,j1)+(GoB f)*(len GoB f,j1+1))+|[1,0]|)
            meets Int cell(GoB f,i1,j1) by A26,A30,A181,A183,A186,GOBOARD6:86;
            LSeg(1/2*((GoB f)*(len GoB f,j2)+(GoB f)*(len GoB f,j1))+|[1,0]|,
            1/2*((GoB f)*(len GoB f,j1)+(GoB f)*(len GoB f,j1+1))+|[1,0]|)
            misses L~f by A28,A30,A54,A181,A182,A186,GOBOARD8:34;
then LSeg(1/2*((GoB f)*(len GoB f,j2)+(GoB f)*(len GoB f,j1))+|[1,0]|,
            1/2*((GoB f)*(len GoB f,j1)+(GoB f)*(len GoB f,j1+1))+|[1,0]|)
            c= (L~f)` by SUBSET_1:23;
            then
            A193:       LSeg
(1/2*((GoB f)*(len GoB f,j2)+(GoB f)*(len GoB f,j1))+|[1,0 ]|,
            1/2*((GoB f)*(len GoB f,j1)+(GoB f)*(len GoB f,j1+1))+|[1,0]|)
            c= LeftComp f by A55,A190,A192,Th4;
A194:         Int left_cell(f,k+1) is convex
              by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A191,A193,A27,A28,A182,A183,Th4,A194,
GOBOARD6:86;
          end;
          suppose that
A195:       i0+1 = i1 and
A196:       j1 = j2+1 and
A197:       i1 <> len GoB f and
A198:       j0 <> width GoB f;
A199:       left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A195,GOBOARD5:28;
            len GoB f > i1 by A21,A197,XXREAL_0:1;
            then
A200:       i1+1 <= len GoB f by NAT_1:13;
            width GoB f > j0 by A25,A198,XXREAL_0:1;
            then
A201:       width GoB f >= j0+1 by NAT_1:13;
A202:       LSeg(1/2*((GoB f)*(i0,j2+1)+(GoB f)*(i0+1,j1+1)),
            1/2*((GoB f)*(i0+1,j2+1)+(GoB f)*(i1+1,j1+1)))
            meets Int cell(GoB f,i0,j1) by A18,A21,A26,A30,A195,A196,A201,
GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i0,j2+1)+(GoB f)*(i0+1,j1+1)),
            1/2*((GoB f)*(i0+1,j2+1)+(GoB f)*(i1+1,j1+1))) misses L~f
by A5,A6,A11,A14,A17,A18,A28,A30,A31,A50,A51,A54,A195,A196,A200,A201,
GOBOARD8:16;
            then LSeg(1/2*((GoB f)*(i0,j2+1)+(GoB f)*(i0+1,j1+1)),
            1/2*((GoB f)*(i0+1,j2+1)+(GoB f)*(i1+1,j1+1)))
            c= (L~f)` by SUBSET_1:23;
            then
A203:       LSeg(1/2*((GoB f)*(i0,j2+1)+(GoB f)*(i0+1,j1+1)),
            1/2*((GoB f)*(i0+1,j2+1)+(GoB f)*(i1+1,j1+1))) c= LeftComp f
            by A3,A5,A6,A55,A199,A202,Th4,NAT_1:13;
A204:         Int cell(GoB f,i1,j1) is convex by A21,A27,Th16;
            Int cell(GoB f,i1,j1) misses L~f by A21,A27,GOBOARD7:12;
            then Int cell(GoB f,i1,j1) c= (L~f)` by SUBSET_1:23;
            then
A205:       Int cell(GoB f,i1,j1) c= LeftComp f by A55,A203,A20,A26,A30,A195
,A196,A200,A201,Th4,A204,GOBOARD6:82;
A206:       left_cell(f,k+1) = cell(GoB f,i1,j2)
            by A4,A5,A13,A14,A16,A17,A31,A45,A196,GOBOARD5:30;
A207:       LSeg(1/2*((GoB f)*(i0+1,j2)+(GoB f)*(i1+1,j2+1)),
            1/2*((GoB f)*(i0+1,j2+1)+(GoB f)*(i1+1,j1+1)))
            meets Int cell(GoB f,i1,j1)
            by A20,A26,A30,A195,A196,A200,A201,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i0+1,j2)+(GoB f)*(i1+1,j2+1)),
            1/2*((GoB f)*(i0+1,j2+1)+(GoB f)*(i1+1,j1+1))) misses L~f
by A5,A6,A11,A14,A17,A18,A28,A30,A31,A50,A51,A54,A195,A196,A200,A201,GOBOARD8:6
            ;
            then LSeg(1/2*((GoB f)*(i0+1,j2)+(GoB f)*(i1+1,j2+1)),
            1/2*((GoB f)*(i0+1,j2+1)+(GoB f)*(i1+1,j1+1))) c= (L~f)`
            by SUBSET_1:23;
            then
A208:       LSeg(1/2*((GoB f)*(i0+1,j2)+(GoB f)*(i1+1,j2+1)),
            1/2*((GoB f)*(i0+1,j2+1)+(GoB f)*(i1+1,j1+1))) c= LeftComp f
            by A55,A205,A207,Th4;
A209:         Int left_cell(f,k+1) is convex
               by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A206,A208,A20,A27,A28,A195,A196,A200,Th4,A209,
GOBOARD6:82;
          end;
          suppose that
A210:       j0+1 = j1 and
A211:       i1 = i2+1;
            left_cell(f,k) = cell(GoB f,i2,j0)
            by A6,A7,A10,A11,A13,A14,A30,A210,A211,GOBOARD5:27
.= left_cell(f,k+1) by A4,A5,A13,A14,A16,A17,A31,A210,A211,GOBOARD5:29;
            hence thesis by A3,A5,A6,NAT_1:13;
          end;
          suppose that
A212:       i0 = i1+1 and
A213:       j1+1 = j2 and
A214:       i1 = 1 and
A215:       j1 = 1;
A216:       left_cell(f,k) = cell(GoB f,i1,j1-'1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A212,GOBOARD5:29;
            i1-'1 <= i1 by NAT_D:35;
            then
A217:       i1-'1 <= len GoB f by A21,XXREAL_0:2;
A218:       j1-'1 = 0 by A215,XREAL_1:232;
A219:       i1-'1 = 0 by A214,XREAL_1:232;
            j1-'1 <= j1 by NAT_D:35;
            then
A220:       j1-'1 <= width GoB f by A27,XXREAL_0:2;
A221:       LSeg((GoB f)*(i1,j1)- |[1,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i0,j1))- |[0,1]|)
meets Int cell(GoB f,i1,j1-'1) by A19,A20,A212,A215,A218,GOBOARD6:84;
            LSeg((GoB f)*(i1,j1)- |[1,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i0,j1))- |[0,1]|)
            misses L~f by A212,A214,A215,GOBOARD8:26;
            then LSeg((GoB f)*(i1,j1)- |[1,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i0,j1))- |[0,1]|) c= (L~f)`
            by SUBSET_1:23;
            then
A222:       LSeg((GoB f)*(i1,j1)- |[1,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i0,j1))- |[0,1]|) c= LeftComp f
            by A3,A5,A6,A55,A216,A221,Th4,NAT_1:13;
A223:         Int cell(GoB f,i1-'1,j1-'1) is convex
            by A217,A220,Th16;
            Int cell(GoB f,i1-'1,j1-'1) misses L~f by A217,A220,GOBOARD7:12;
            then Int cell(GoB f,i1-'1,j1-'1) c= (L~f)` by SUBSET_1:23;
            then
A224:       Int cell(GoB f,i1-'1,j1-'1) c= LeftComp f
            by A55,A222,A214,A215,A218,Th4,A223,GOBOARD6:87;
A225:       left_cell(f,k+1) = cell(GoB f,i1-'1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A45,A213,GOBOARD5:27;
A226:       LSeg((GoB f)*(i1,j1)- |[1,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1,j2))- |[1,0]|)
            meets Int cell(GoB f,i1-'1,j1-'1) by A214,A215,A218,GOBOARD6:87;
            LSeg((GoB f)*(i1,j1)- |[1,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1,j2))- |[1,0]|)
            misses L~f by A213,A214,A215,GOBOARD8:32;
            then LSeg((GoB f)*(i1,j1)- |[1,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1,j2))- |[1,0]|) c= (L~f)`
            by SUBSET_1:23;
            then
A227:       LSeg((GoB f)*(i1,j1)- |[1,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i1,j2))- |[1,0]|) c= LeftComp f
            by A55,A224,A226,Th4;
A228:         Int left_cell(f,k+1) is convex
               by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A225,A227,A26,A29,A213,A214,A219,Th4,A228,
GOBOARD6:85;
          end;
          suppose that
A229:       i0 = i1+1 and
A230:       j1+1 = j2 and
A231:       i1 <> 1 and
A232:       j1 = 1;
A233:       left_cell(f,k) = cell(GoB f,i1,j1-'1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A229,GOBOARD5:29;
            1 < i1 by A20,A231,XXREAL_0:1;
            then
A234:       1 <= i1-'1 by A45,NAT_1:13;
A235:       i1-'1+(1+1) = i0 by A45,A229;
            i1-'1 <= i1 by NAT_D:35;
            then
A236:       i1-'1 <= len GoB f by A21,XXREAL_0:2;
A237:       j1-'1 = 0 by A232,XREAL_1:232;
            j1-'1 <= j1 by NAT_D:35;
            then
A238:       j1-'1 <= width GoB f by A27,XXREAL_0:2;
A239:       LSeg(1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,1))- |[0,1]|,
            1/2*((GoB f)*(i1,1)+(GoB f)*(i0,1))- |[0,1]|)
            meets Int cell(GoB f,i1,j1-'1) by A19,A20,A229,A237,GOBOARD6:84;
            LSeg(1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,1))- |[0,1]|,
            1/2*((GoB f)*(i1,1)+(GoB f)*(i0,1))- |[0,1]|) misses L~f
            by A19,A45,A234,A235,GOBOARD8:25;
            then LSeg(1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,1))- |[0,1]|,
            1/2*((GoB f)*(i1,1)+(GoB f)*
            (i0,1))- |[0,1]|) c= (L~f)` by SUBSET_1:23;
            then
A240:       LSeg(1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,1))- |[0,1 ]|,
            1/2*((GoB f)*(i1,1)+(GoB f)*(i0,1))- |[0,1]|) c= LeftComp f
            by A3,A5,A6,A55,A233,A239,Th4,NAT_1:13;
A241:       Int cell(GoB f,i1-'1,j1-'1) is convex
            by A236,A238,Th16;
            Int cell(GoB f,i1-'1,j1-'1) misses L~f by A236,A238,GOBOARD7:12;
            then Int cell(GoB f,i1-'1,j1-'1) c= (L~f)` by SUBSET_1:23;
            then
A242:       Int cell(GoB f,i1-'1,j1-'1) c= LeftComp f
            by A55,A240,A21,A45,A234,A237,Th4,A241,GOBOARD6:84;
A243:       left_cell(f,k+1) = cell(GoB f,i1-'1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A45,A230,GOBOARD5:27;
A244:       LSeg(1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,1))- |[0,1]|,
            1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,2)))
            meets Int cell(GoB f,i1-'1,j1-'1) by A21,A45,A234,A237,GOBOARD6:84;
            LSeg(1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,1))- |[0,1]|,
            1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,2))) misses L~f
            by A5,A6,A11,A14,A17,A19,A30,A31,A45,A50,A230,A232,A234,A235,
GOBOARD8:7;
            then LSeg(1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,1))- |[0,1]|,
            1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,2))) c= (L~f)` by SUBSET_1:23;
            then
A245:       LSeg(1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,1))- |[0,1 ]|,
            1/2*((GoB f)*(i1-'1,1)+(GoB f)*(i1,2))) c= LeftComp f
            by A55,A242,A244,Th4;
A246:       Int left_cell(f,k+1) is convex
               by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A243,A245,A21,A29,A45,A230,A232,A234,Th4,A246,
GOBOARD6:82;
          end;
          suppose that
A247:       i0 = i1+1 and
A248:       j1+1 = j2 and
A249:       i1 = 1 and
A250:       j1 <> 1;
A251:       left_cell(f,k) = cell(GoB f,i1,j1-'1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A247,GOBOARD5:29;
            1 < j0 by A24,A30,A247,A250,XXREAL_0:1;
            then
A252:       1 <= j0-'1 by A30,A46,A247,NAT_1:13;
A253:       j0-'1+(1+1) = j2 by A30,A46,A247,A248;
A254:       LSeg(1/2*((GoB f)*(1,j1-'1)+(GoB f)*(1,j1-'1+1))- |[1,0]|,
            1/2*((GoB f)*(1,j1-'1)+(GoB f)*(2,j1-'1+1)))
            meets Int cell(GoB f,i1,j1-'1)
            by A19,A27,A30,A46,A247,A249,A252,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(1,j1-'1)+(GoB f)*(1,j1-'1+1))- |[1,0]|,
            1/2*((GoB f)*(1,j1-'1)+(GoB f)*(2,j1-'1+1))) misses L~f
            by A5,A6,A11,A14,A17,A29,A30,A31,A46,A50,A247,A248,A249,A252,A253,
GOBOARD8:17;
            then LSeg(1/2*((GoB f)*(1,j1-'1)+(GoB f)*(1,j1-'1+1))- |[1,0]|,
            1/2*((GoB f)*(1,j1-'1)+(GoB f)*(2,j1-'1+1))) c= (L~f)`
            by SUBSET_1:23;
            then
            A255:       LSeg
(1/2*((GoB f)*(1,j1-'1)+(GoB f)*(1,j1-'1+1))- |[1,0]|,
            1/2*((GoB f)*(1,j1-'1)+(GoB f)*(2,j1-'1+1))) c= LeftComp f
            by A3,A5,A6,A55,A251,A254,Th4,NAT_1:13;
            i1-'1 <= i1 by NAT_D:35;
            then
A256:       i1-'1 <= len GoB f by A21,XXREAL_0:2;
A257:       i1-'1 = 0 by A249,XREAL_1:232;
            j1-'1 <= j1 by NAT_D:35;
            then
A258:       j1-'1 <= width GoB f by A27,XXREAL_0:2;
A259:         Int cell(GoB f,i1-'1,j1-'1) is convex
            by A256,A258,Th16;
            Int cell(GoB f,i1-'1,j1-'1) misses L~f by A256,A258,GOBOARD7:12;
            then Int cell(GoB f,i1-'1,j1-'1) c= (L~f)` by SUBSET_1:23;
            then
A260:       Int cell(GoB f,i1-'1,j1-'1) c= LeftComp f
            by A30,A55,A247,A255,A27,A46,A252,A257,Th4,A259,GOBOARD6:85;
A261:       left_cell(f,k+1) = cell(GoB f,i1-'1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A45,A248,GOBOARD5:27;
A262:       LSeg(1/2*((GoB f)*(1,j1-'1)+(GoB f)*(1,j1-'1+1))- |[1,0]|,
            1/2*((GoB f)*(1,j1-'1+1)+(GoB f)*(1,j1-'1+2))- |[1,0]|)
            meets Int cell(GoB f,i1-'1,j1-'1)
            by A27,A30,A46,A247,A252,A257,GOBOARD6:85;
            LSeg(1/2*((GoB f)*(1,j1-'1)+(GoB f)*(1,j1-'1+1))- |[1,0]|,
            1/2*((GoB f)*(1,j1-'1+1)+(GoB f)*(1,j1-'1+2))- |[1,0]|) misses L~f
            by A29,A30,A46,A247,A248,A252,GOBOARD8:31;
            then LSeg(1/2*((GoB f)*(1,j1-'1)+(GoB f)*(1,j1-'1+1))- |[1,0]|,
            1/2*((GoB f)*(1,j1-'1+1)+(GoB f)*(1,j1-'1+2))- |[1,0]|) c= (L~f)`
            by SUBSET_1:23;
            then
            A263:       LSeg
(1/2*((GoB f)*(1,j1-'1)+(GoB f)*(1,j1-'1+1))- |[1,0]|,
1/2*((GoB f)*(1,j1-'1+1)+(GoB f)*(1,j1-'1+2))- |[1,0]|) c= LeftComp f
            by A55,A260,A262,Th4;
A264:         Int left_cell(f,k+1) is convex
                by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A261,A263,A26,A29,A46,A248,A257,Th4,A264,
GOBOARD6:85;
          end;
          suppose that
A265:       i0 = i1+1 and
A266:       j1+1 = j2 and
A267:       i1 <> 1 and
A268:       j1 <> 1;
A269:       left_cell(f,k) = cell(GoB f,i1,j1-'1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A265,GOBOARD5:29;
            1 < j0 by A24,A30,A265,A268,XXREAL_0:1;
            then
A270:       1 <= j0-'1 by A30,A46,A265,NAT_1:13;
            1 < i1 by A20,A267,XXREAL_0:1;
            then
A271:       1 <= i1-'1 by A45,NAT_1:13;
A272:       i1-'1+(1+1) = i0 by A45,A265;
A273:       j0-'1+(1+1) = j2 by A30,A46,A265,A266;
A274:       LSeg(1/2*((GoB f)*(i1-'1,j0-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1,j0-'1)+(GoB f)*(i0,j1)))
            meets Int cell(GoB f,i1,j1-'1)
            by A19,A20,A27,A30,A46,A265,A270,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i1-'1,j0-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1,j0-'1)+(GoB f)*(i0,j1))) misses L~f
by A5,A6,A11,A14,A17,A19,A29,A30,A31,A45,A46,A50,A266,A270,A271,A272,A273,
GOBOARD8:12;
            then LSeg(1/2*((GoB f)*(i1-'1,j0-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1,j0-'1)+(GoB f)*(i0,j1))) c= (L~f)` by SUBSET_1:23;
            then
A275:       LSeg(1/2*((GoB f)*(i1-'1,j0-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1,j0-'1)+(GoB f)*(i0,j1))) c= LeftComp f
            by A3,A5,A6,A55,A269,A274,Th4,NAT_1:13;
            i1-'1 <= i1 by NAT_D:35;
            then
A276:       i1-'1 <= len GoB f by A21,XXREAL_0:2;
            j1-'1 <= j1 by NAT_D:35;
            then
A277:       j1-'1 <= width GoB f by A27,XXREAL_0:2;
A278:         Int cell(GoB f,i1-'1,j1-'1) is convex
            by A276,A277,Th16;
            Int cell(GoB f,i1-'1,j1-'1) misses L~f by A276,A277,GOBOARD7:12;
            then Int cell(GoB f,i1-'1,j1-'1) c= (L~f)` by SUBSET_1:23;
            then
A279:       Int cell(GoB f,i1-'1,j1-'1) c= LeftComp f
            by A30,A55,A265,A275,A21,A27,A45,A46,A270,A271,Th4,A278,GOBOARD6:82
;
A280:       left_cell(f,k+1) = cell(GoB f,i1-'1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A45,A266,GOBOARD5:27;
A281:       LSeg(1/2*((GoB f)*(i1-'1,j1-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j2)))
            meets Int cell(GoB f,i1-'1,j1-'1)
            by A21,A27,A30,A45,A46,A265,A270,A271,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i1-'1,j1-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j2))) misses L~f
by A5,A6,A11,A14,A17,A19,A29,A30,A31,A45,A46,A50,A266,A270,A271,A272,A273,
GOBOARD8:2;
            then LSeg(1/2*((GoB f)*(i1-'1,j1-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j2))) c= (L~f)` by SUBSET_1:23;
            then
A282:       LSeg(1/2*((GoB f)*(i1-'1,j1-'1)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j2))) c= LeftComp f
            by A55,A279,A281,Th4;
A283:         Int left_cell(f,k+1) is convex
                by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A280,A282,A21,A26,A29,A45,A266,A271,Th4,A283,
GOBOARD6:82;
          end;
          suppose that
A284:       j0 = j1+1 and
A285:       i1+1 = i2;
            left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A45,A284,GOBOARD5:30
              .= left_cell(f,k+1)
            by A4,A5,A13,A14,A16,A17,A30,A31,A46,A284,A285,GOBOARD5:28;
            hence thesis by A3,A5,A6,NAT_1:13;
          end;
          suppose that
A286:       i0+1 = i1 and
A287:       i1+1 = i2 and
A288:       j0 = width GoB f;
A289:       left_cell(f,k) = cell(GoB f,i0,width GoB f)
            by A6,A7,A10,A11,A13,A14,A30,A49,A286,A288,GOBOARD5:28;
A290:       left_cell(f,k+1) = cell(GoB f,i1,width GoB f)
            by A4,A5,A13,A14,A16,A17,A30,A31,A49,A286,A287,A288,GOBOARD5:28;
A291:       LSeg
            (1/2*((GoB f)*(i0,width GoB f)+(GoB f)*(i0+1,width GoB f))+|[0,1]|,
            1/2*((GoB f)*(i1,width GoB f)+(GoB f)*(i1+1,width GoB f))+|[0,1]|)
            meets Int cell(GoB f,i0,width GoB f) by A18,A21,A286,GOBOARD6:83;
LSeg(1/2*((GoB f)*(i0,width GoB f)+(GoB f)*(i0+1,width GoB f))+|[0,1 ]| ,
            1/2*((GoB f)*(i1,width GoB f)+(GoB f)*(i1+1,width GoB f))+|[0,1]|)
            misses L~f by A18,A23,A51,A286,A287,GOBOARD8:28;
then LSeg(1/2*((GoB f)*(i0,width GoB f)+(GoB f)*(i0+1,width GoB f))+|[0,1 ]|,
            1/2*((GoB f)*(i1,width GoB f)+(GoB f)*(i1+1,width GoB f))+|[0,1]|)
            c= (L~f)` by SUBSET_1:23;
            then
            A292:       LSeg
(1/2*((GoB f)*(i0,width GoB f)+(GoB f)*(i0+1, width GoB f))+
            |[0,1]|,
            1/2*((GoB f)*(i1,width GoB f)+(GoB f)*(i1+1,width GoB f))+|[0,1]|)
            c= LeftComp f by A3,A5,A6,A55,A289,A291,Th4,NAT_1:13;
A293:         Int left_cell(f,k+1) is convex
                by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A290,A292,A20,A23,A287,Th4,A293,GOBOARD6:83;
          end;
          suppose that
A294:       i0+1 = i1 and
A295:       i1+1 = i2 and
A296:       j0 <> width GoB f;
A297:       left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A294,GOBOARD5:28;
            width GoB f > j0 by A25,A296,XXREAL_0:1;
            then
A298:       width GoB f >= j0+1 by NAT_1:13;
A299:       left_cell(f,k+1) = cell(GoB f,i1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A46,A295,GOBOARD5:28;
A300:       LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0+1,j0+1)),
            1/2*((GoB f)*(i1,j0)+(GoB f)*(i1+1,j0+1)))
            meets Int cell(GoB f,i0,j1)
            by A18,A21,A26,A30,A294,A298,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0+1,j0+1)),
            1/2*((GoB f)*(i1,j0)+(GoB f)*(i1+1,j0+1))) misses L~f
            by A5,A6,A11,A14,A17,A18,A23,A24,A30,A31,A50,A51,A294,A295,A298,
GOBOARD8:14;
            then LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0+1,j0+1)),
            1/2*((GoB f)*(i1,j0)+(GoB f)*
            (i1+1,j0+1))) c= (L~f)` by SUBSET_1:23;
            then
A301:       LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0+1,j0+1)),
            1/2*((GoB f)*(i1,j0)+(GoB f)*(i1+1,j0+1))) c= LeftComp f
            by A3,A5,A6,A55,A297,A300,Th4,NAT_1:13;
A302:         Int left_cell(f,k+1) is convex
              by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A299,A301,A20,A23,A24,A30,A294,A295,A298,Th4
,A302,GOBOARD6:82;
          end;
          suppose that
A303:       i0+1 = i1 and
A304:       j1+1 = j2;
            left_cell(f,k) = cell(GoB f,i0,j1)
            by A6,A7,A10,A11,A13,A14,A30,A46,A303,GOBOARD5:28
.= left_cell(f,k+1) by A4,A5,A13,A14,A16,A17,A31,A303,A304,GOBOARD5:27;
            hence thesis by A3,A5,A6,NAT_1:13;
          end;
          suppose that
A305:       j0+1 = j1 and
A306:       i1+1 = i2 and
A307:       i0 = 1 and
A308:       j1 = width GoB f;
A309:       left_cell(f,k) = cell(GoB f,i1-'1,j0)
            by A6,A7,A10,A11,A13,A14,A30,A45,A305,GOBOARD5:27;
A310:       i0-'1 = 0 by A307,XREAL_1:232;
A311:       j1-'1 = j0 by A305,NAT_D:34;
A312:       LSeg(1/2*((GoB f)*(1,j0)+(GoB f)*(1,j1))- |[1,0]|,
            (GoB f)*(1,j1)+|[-1,1]|) meets Int cell(GoB f,i1-'1,j0)
            by A24,A27,A30,A305,A310,GOBOARD6:85;
            LSeg(1/2*((GoB f)*(1,j0)+(GoB f)*(1,j1))- |[1,0]|,
            (GoB f)*(1,j1)+|[-1,1]|) misses L~f by A308,A311,GOBOARD8:33;
            then LSeg(1/2*((GoB f)*(1,j0)+(GoB f)*(1,j1))- |[1,0]|,
            (GoB f)*(1,j1)+|[-1,1]|) c= (L~f)` by SUBSET_1:23;
            then
A313:       LSeg(1/2*((GoB f)*(1,j0)+(GoB f)*(1,j1))- |[1,0]|,
            (GoB f)*(1,j1)+|[-1,1]|) c= LeftComp f
            by A3,A5,A6,A55,A309,A312,Th4,NAT_1:13;
            i1-'1 <= i1 by NAT_D:35;
            then
A314:       i1-'1 <= len GoB f by A21,XXREAL_0:2;
A315:         Int cell(GoB f,i1-'1,j1) is convex by A27,A314,Th16;
            Int cell(GoB f,i1-'1,j1) misses L~f by A27,A314,GOBOARD7:12;
            then Int cell(GoB f,i1-'1,j1) c= (L~f)` by SUBSET_1:23;
            then
A316:       Int cell(GoB f,i1-'1,j1) c= LeftComp f by A30,A55,A305,A313,A308
,A310,Th4,A315,GOBOARD6:89;
A317:       left_cell(f,k+1) = cell(GoB f,i1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A305,A306,GOBOARD5:28;
A318:       LSeg((GoB f)*(1,j1)+|[-1,1]|,
            1/2*((GoB f)*(1,j1)+(GoB f)*(2,j1))+|[0,1]|)
            meets Int cell(GoB f,i1-'1,j1) by A30,A305,A308,A310,GOBOARD6:89;
            LSeg((GoB f)*(1,j1)+|[-1,1]|,
            1/2*((GoB f)*(1,j1)+(GoB f)*(2,j1))+|[0,1]|) misses L~f
            by A308,GOBOARD8:29;
            then LSeg((GoB f)*(1,j1)+|[-1,1]|,
            1/2*((GoB f)*(1,j1)+(GoB f)*(2,j1))+|[0,1]|) c= (L~f)`
            by SUBSET_1:23;
            then
A319:       LSeg((GoB f)*(1,j1)+|[-1,1]|,
            1/2*((GoB f)*(1,j1)+(GoB f)*(2,j1))+|[0,1]|) c= LeftComp f
            by A55,A316,A318,Th4;
A320:         Int left_cell(f,k+1) is convex
                by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A317,A319,A23,A30,A305,A306,A307,A308,Th4,A320,
GOBOARD6:83;
          end;
          suppose that
A321:       j0+1 = j1 and
A322:       i1+1 = i2 and
A323:       i0 <> 1 and
A324:       j1 = width GoB f;
A325:       left_cell(f,k) = cell(GoB f,i1-'1,j0)
            by A6,A7,A10,A11,A13,A14,A30,A45,A321,GOBOARD5:27;
            1 < i0 by A18,A323,XXREAL_0:1;
            then
A326:       1 <= i0-'1 by A30,A45,A321,NAT_1:13;
A327:       j1-'1 = j0 by A321,NAT_D:34;
A328:       i1-'1+(1+1) = i2 by A45,A322;
A329:       LSeg(1/2*((GoB f)*(i1-'1,j0)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1))+|[0,1]|)
            meets Int cell(GoB f,i1-'1,j0)
            by A21,A24,A27,A30,A45,A321,A326,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i1-'1,j0)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1))+|[0,1]|) misses L~f
            by A5,A6,A11,A14,A17,A23,A30,A31,A45,A50,A321,A324,A326,A327,A328,
GOBOARD8:9;
            then LSeg(1/2*((GoB f)*(i1-'1,j0)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1))+|[0,1]|) c= (L~f)`
            by SUBSET_1:23;
            then
A330:       LSeg(1/2*((GoB f)*(i1-'1,j0)+(GoB f)*(i1,j1)),
            1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1))+|[0,1]|) c= LeftComp f
            by A3,A5,A6,A55,A325,A329,Th4,NAT_1:13;
            i1-'1 <= i1 by NAT_D:35;
            then
A331:       i1-'1 <= len GoB f by A21,XXREAL_0:2;
A332:         Int cell(GoB f,i1-'1,j1) is convex by A27,A331,Th16;
            Int cell(GoB f,i1-'1,j1) misses L~f by A27,A331,GOBOARD7:12;
            then Int cell(GoB f,i1-'1,j1) c= (L~f)` by SUBSET_1:23;
            then
A333:       Int cell(GoB f,i1-'1,j1) c= LeftComp f by A30,A55,A321,A330,A19,A45
,A324,A326,Th4,A332,GOBOARD6:83;
A334:       left_cell(f,k+1) = cell(GoB f,i1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A321,A322,GOBOARD5:28;
A335:       LSeg(1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1))+|[0,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i2,j1))+|[0,1]|)
            meets Int cell(GoB f,i1-'1,j1) by A21,A30,A45,A321,A324,A326,
GOBOARD6:83;
            LSeg(1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1))+|[0,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i2,j1))+|[0,1]|) misses L~f
            by A23,A30,A45,A321,A324,A326,A328,GOBOARD8:28;
            then LSeg(1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1))+|[0,1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i2,j1))+|[0,1]|) c= (L~f)`
            by SUBSET_1:23;
            then
A336:       LSeg(1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1))+|[0, 1]|,
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i2,j1))+|[0,1]|) c= LeftComp f
            by A55,A333,A335,Th4;
A337:         Int left_cell(f,k+1) is convex
                 by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A334,A336,A20,A23,A322,A324,Th4,A337,
GOBOARD6:83;
          end;
          suppose that
A338:       j0+1 = j1 and
A339:       i1+1 = i2 and
A340:       i0 = 1 and
A341:       j1 <> width GoB f;
A342:       left_cell(f,k) = cell(GoB f,i1-'1,j0)
            by A6,A7,A10,A11,A13,A14,A30,A45,A338,GOBOARD5:27;
            width GoB f > j1 by A27,A341,XXREAL_0:1;
            then
A343:       width GoB f >= j1+1 by NAT_1:13;
A344:       i0-'1 = 0 by A340,XREAL_1:232;
A345:       LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0,j1))- |[1,0]|,
            1/2*((GoB f)*(i0,j1)+(GoB f)*(i0,j1+1))- |[1,0]|)
            meets Int cell(GoB f,i1-'1,j0)
            by A24,A27,A30,A338,A340,A344,GOBOARD6:85;
            LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0,j1))- |[1,0]|,
            1/2*((GoB f)*(i0,j1)+(GoB f)*(i0,j1+1))- |[1,0]|) misses L~f
            by A24,A53,A338,A340,A343,GOBOARD8:31;
            then LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0,j1))- |[1,0]|,
            1/2*((GoB f)*(i0,j1)+(GoB f)*(i0,j1+1))- |[1,0]|) c= (L~f)`
            by SUBSET_1:23;
            then
A346:       LSeg(1/2*((GoB f)*(i0,j0)+(GoB f)*(i0,j1))- |[1,0 ]|,
            1/2*((GoB f)*(i0,j1)+(GoB f)*(i0,j1+1))- |[1,0]|) c= LeftComp f
            by A3,A5,A6,A55,A342,A345,Th4,NAT_1:13;
            i1-'1 <= i1 by NAT_D:35;
            then
A347:       i1-'1 <= len GoB f by A21,XXREAL_0:2;
A348:         Int cell(GoB f,i1-'1,j1) is convex by A27,A347,Th16;
            Int cell(GoB f,i1-'1,j1) misses L~f by A27,A347,GOBOARD7:12;
            then Int cell(GoB f,i1-'1,j1) c= (L~f)` by SUBSET_1:23;
            then
A349:       Int cell(GoB f,i1-'1,j1) c= LeftComp f by A30,A55,A338,A346,A26
,A340,A343,A344,Th4,A348,GOBOARD6:85;
A350:       left_cell(f,k+1) = cell(GoB f,i1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A338,A339,GOBOARD5:28;
A351:       LSeg(1/2*((GoB f)*(i1,j0+1)+(GoB f)*(i1,j0+2))- |[1,0]|,
            1/2*((GoB f)*(i1,j0+1)+(GoB f)*(i2,j0+2)))
            meets Int cell(GoB f,i1-'1,j1)
            by A26,A30,A338,A340,A343,A344,GOBOARD6:85;
            LSeg(1/2*((GoB f)*(i1,j0+1)+(GoB f)*(i1,j0+2))- |[1,0]|,
            1/2*((GoB f)*(i1,j0+1)+(GoB f)*(i2,j0+2))) misses L~f
by A5,A6,A11,A14,A17,A24,A30,A31,A50,A338,A339,A340,A343,GOBOARD8:18;
            then LSeg(1/2*((GoB f)*(i1,j0+1)+(GoB f)*(i1,j0+2))- |[1,0]|,
            1/2*((GoB f)*(i1,j0+1)+(GoB f)*
            (i2,j0+2))) c= (L~f)` by SUBSET_1:23;
            then
            A352:       LSeg
(1/2*((GoB f)*(i1,j0+1)+(GoB f)*(i1,j0+2))- |[ 1,0]|,
            1/2*((GoB f)*(i1,j0+1)+(GoB f)*(i2,j0+2))) c= LeftComp f
            by A55,A349,A351,Th4;
A353:         Int left_cell(f,k+1) is convex
               by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A350,A352,A20,A23,A26,A338,A339,A343,Th4,A353,
GOBOARD6:82;
          end;
          suppose that
A354:       j0+1 = j1 and
A355:       i1+1 = i2 and
A356:       i0 <> 1 and
A357:       j1 <> width GoB f;
A358:       left_cell(f,k) = cell(GoB f,i1-'1,j0)
            by A6,A7,A10,A11,A13,A14,A30,A45,A354,GOBOARD5:27;
            1 < i0 by A18,A356,XXREAL_0:1;
            then
A359:       1 <= i0-'1 by A30,A45,A354,NAT_1:13;
            width GoB f > j1 by A27,A357,XXREAL_0:1;
            then
A360:       width GoB f >= j1+1 by NAT_1:13;
A361:       i1-'1+(1+1) = i2 by A45,A355;
A362:       LSeg(1/2*((GoB f)*(i0-'1,j0)+(GoB f)*(i0,j1)),
            1/2*((GoB f)*(i0-'1,j1)+(GoB f)*(i0,j1+1)))
            meets Int cell(GoB f,i1-'1,j0)
            by A21,A24,A27,A30,A45,A354,A359,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i0-'1,j0)+(GoB f)*(i0,j1)),
            1/2*((GoB f)*(i0-'1,j1)+(GoB f)*(i0,j1+1))) misses L~f
                by A5,A6,A11,A14,A17,A23,A24,A30,A31,A45,A50,A53,A354,A359,A360
,A361,GOBOARD8:3;
            then LSeg(1/2*((GoB f)*(i0-'1,j0)+(GoB f)*(i0,j1)),
            1/2*((GoB f)*(i0-'1,j1)+(GoB f)*
            (i0,j1+1))) c= (L~f)` by SUBSET_1:23;
            then
A363:       LSeg(1/2*((GoB f)*(i0-'1,j0)+(GoB f)*(i0,j1)),
            1/2*((GoB f)*(i0-'1,j1)+(GoB f)*(i0,j1+1))) c= LeftComp f
            by A3,A5,A6,A55,A358,A362,Th4,NAT_1:13;
            i1-'1 <= i1 by NAT_D:35;
            then
A364:       i1-'1 <= len GoB f by A21,XXREAL_0:2;
A365:         Int cell(GoB f,i1-'1,j1) is convex by A27,A364,Th16;
            Int cell(GoB f,i1-'1,j1) misses L~f by A27,A364,GOBOARD7:12;
            then Int cell(GoB f,i1-'1,j1) c= (L~f)` by SUBSET_1:23;
            then
A366:       Int cell(GoB f,i1-'1,j1) c= LeftComp f by A30,A55,A354,A363,A19,A26
,A45,A359,A360,Th4,A365,GOBOARD6:82;
A367:       left_cell(f,k+1) = cell(GoB f,i1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A354,A355,GOBOARD5:28;
A368:       LSeg(1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1+1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i2,j1+1)))
            meets Int cell(GoB f,i1-'1,j1) by A21,A26,A30,A45,A354,A359,A360,
GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1+1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i2,j1+1))) misses L~f
by A5,A6,A11,A14,A17,A23,A24,A30,A31,A45,A50,A53,A354,A359,A360,A361,
GOBOARD8:15;
            then LSeg(1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1+1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i2,j1+1))) c= (L~f)` by SUBSET_1:23;
            then
A369:       LSeg(1/2*((GoB f)*(i1-'1,j1)+(GoB f)*(i1,j1+1)),
            1/2*((GoB f)*(i1,j1)+(GoB f)*(i2,j1+1))) c= LeftComp f
            by A55,A366,A368,Th4;
A370:         Int left_cell(f,k+1) is convex
               by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A367,A369,A20,A23,A26,A355,A360,Th4,A370,
GOBOARD6:82;
          end;
          suppose that
A371:       j0+1 = j1 and
A372:       j1+1 = j2 and
A373:       i0 = 1;
A374:       left_cell(f,k) = cell(GoB f,0,j0)
            by A6,A7,A10,A11,A13,A14,A30,A56,A371,A373,GOBOARD5:27;
A375:       left_cell(f,k+1) = cell(GoB f,0,j1)
            by A4,A5,A13,A14,A16,A17,A30,A31,A56,A371,A372,A373,GOBOARD5:27;
A376:       LSeg(1/2*((GoB f)*(1,j0)+(GoB f)*(1,j1))- |[1,0]|,
            1/2*((GoB f)*(1,j1)+(GoB f)*(1,j2))- |[1,0]|)
            meets Int cell(GoB f,0,j0) by A24,A27,A371,GOBOARD6:85;
            LSeg(1/2*((GoB f)*(1,j0)+(GoB f)*(1,j1))- |[1,0]|,
            1/2*((GoB f)*(1,j1)+(GoB f)*(1,j2))- |[1,0]|) misses L~f
            by A24,A29,A53,A371,A372,GOBOARD8:31;
            then LSeg(1/2*((GoB f)*(1,j0)+(GoB f)*(1,j1))- |[1,0]|,
            1/2*((GoB f)*(1,j1)+(GoB f)*(1,j2))- |[1,0]|) c= (L~f)`
            by SUBSET_1:23;
            then
A377:       LSeg(1/2*((GoB f)*(1,j0)+(GoB f)*(1,j1))- |[1,0]|,
            1/2*((GoB f)*(1,j1)+(GoB f)*(1,j2))- |[1,0]|) c= LeftComp f
            by A3,A5,A6,A55,A374,A376,Th4,NAT_1:13;
A378:         Int left_cell(f,k+1) is convex
               by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A55,A375,A377,A26,A29,A372,Th4,A378,GOBOARD6:85;
          end;
          suppose that
A379:       j0+1 = j1 and
A380:       j1+1 = j2 and
A381:       i0 <> 1;
A382:       left_cell(f,k) = cell(GoB f,i1-'1,j0)
            by A6,A7,A10,A11,A13,A14,A30,A45,A379,GOBOARD5:27;
            1 < i0 by A18,A381,XXREAL_0:1;
            then
A383:       1 <= i0-'1 by A30,A45,A379,NAT_1:13;
A384:       left_cell(f,k+1) = cell(GoB f,i1-'1,j1)
            by A4,A5,A13,A14,A16,A17,A31,A45,A380,GOBOARD5:27;
A385:       LSeg(1/2*((GoB f)*(i0-'1,j0)+(GoB f)*(i0,j1)),
            1/2*((GoB f)*(i0-'1,j1)+(GoB f)*(i0,j2)))
            meets Int cell(GoB f,i1-'1,j0)
            by A21,A24,A27,A30,A45,A379,A383,GOBOARD6:82;
            LSeg(1/2*((GoB f)*(i0-'1,j0)+(GoB f)*(i0,j1)),
            1/2*((GoB f)*(i0-'1,j1)+(GoB f)*(i0,j2))) misses L~f
by A5,A6,A11,A14,A17,A19,A24,A29,A30,A31,A45,A50,A53,A379,A380,A383,GOBOARD8:1;
            then LSeg(1/2*((GoB f)*(i0-'1,j0)+(GoB f)*(i0,j1)),
            1/2*((GoB f)*(i0-'1,j1)+(GoB f)*(i0,j2))) c= (L~f)` by SUBSET_1:23;
            then
A386:       LSeg(1/2*((GoB f)*(i0-'1,j0)+(GoB f)*(i0,j1)),
            1/2*((GoB f)*(i0-'1,j1)+(GoB f)*(i0,j2))) c= LeftComp f
            by A3,A5,A6,A55,A382,A385,Th4,NAT_1:13;
A387:         Int left_cell(f,k+1) is convex
              by A4,A5,Th17;
            Int left_cell(f,k+1) misses L~f by A4,A5,GOBOARD8:37;
            then Int left_cell(f,k+1) c= (L~f)` by SUBSET_1:23;
            hence thesis by A30,A55,A379,A384,A386,A19,A26,A29,A45,A380,A383
,Th4,A387,GOBOARD6:82;
          end;
        end;
        hence thesis;
      end;
    end;
  end;
  thus for k holds P[k] from NAT_1:sch 2(A1,A2);
end;
