reserve i,i1,i2,i9,i19,j,j1,j2,j9,j19,k,k1,k2,l,m,n for Nat;
reserve r,s,r9,s9 for Real;
reserve D for non empty set, f for FinSequence of D;
reserve f for FinSequence of TOP-REAL 2, G for Go-board;

theorem Th11:
  for G1,G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in
Indices G1 & [i2,j2] in Indices G2 & G1*(i1,j1) = G2*(i2,j2) holds cell(G2,i2-'
  1,j2) c= cell(G1,i1-'1,j1)
proof
  let G1,G2 be Go-board such that
A1: Values G1 c= Values G2 and
A2: [i1,j1] in Indices G1 and
A3: [i2,j2] in Indices G2 and
A4: G1*(i1,j1) = G2*(i2,j2);
A5: i2 <= len G2 by A3,MATRIX_0:32;
A6: j1 <= width G1 by A2,MATRIX_0:32;
A7: 1 <= j1 by A2,MATRIX_0:32;
A8: j2 <= width G2 by A3,MATRIX_0:32;
A9: 1 <= j2 by A3,MATRIX_0:32;
A10: 1 <= i2 by A3,MATRIX_0:32;
  then
A11: G2*(i2,j2)`1 = G2*(i2,1)`1 by A5,A9,A8,GOBOARD5:2;
A12: G2*(i2,j2)`2 = G2*(1,j2)`2 by A10,A5,A9,A8,GOBOARD5:1;
  let p be object such that
A13: p in cell(G2,i2-'1,j2);
A14: 1 <= i1 by A2,MATRIX_0:32;
A15: i1 <= len G1 by A2,MATRIX_0:32;
  per cases by A14,A10,XXREAL_0:1;
  suppose
A16: i1 = 1 & i2 = 1;
    then
A17: i1-'1 = 0 by XREAL_1:232;
A18: i2-'1 = 0 by A16,XREAL_1:232;
    now
      per cases by A8,XXREAL_0:1;
      suppose
A19:    j2 = width G2;
        then
A20:    j1 = width G1 by A1,A2,A4,A10,A5,Th5;
        p in { |[r,s]| : r <= G2*(1,1)`1 & G2*(1,width G2)`2 <= s } by A13,A18
,A19,GOBRD11:25;
        then consider r9,s9 such that
A21:    p = |[r9,s9]| and
A22:    r9 <= G2*(1,1)`1 and
A23:    G2*(1,width G2)`2 <= s9;
        G2*(1,1)`1 = G2*(i1,j2)`1 by A5,A9,A8,A16,GOBOARD5:2;
        then r9 <= G1*(1,1)`1 by A4,A15,A7,A6,A16,A22,GOBOARD5:2;
        then
        p in { |[r,s]| : r <= G1*(1,1)`1 & G1*(1,width G1)`2 <= s } by A4,A16
,A19,A21,A23,A20;
        hence thesis by A17,A20,GOBRD11:25;
      end;
      suppose
A24:    j2 < width G2;
        then
        p in {|[r,s]|: r <= G2*(1,1)`1 & G2*(1,j2)`2 <= s & s <= G2*(1,j2
        +1)`2} by A13,A9,A18,GOBRD11:26;
        then consider r9,s9 such that
A25:    p = |[r9,s9]| and
A26:    r9 <= G2*(1,1)`1 and
A27:    G2*(1,j2)`2 <= s9 and
A28:    s9 <= G2*(1,j2+1)`2;
        G2*(1,1)`1 = G2*(i1,j2)`1 by A5,A9,A8,A16,GOBOARD5:2;
        then
A29:    r9 <= G1*(1,1)`1 by A4,A15,A7,A6,A16,A26,GOBOARD5:2;
        now
          per cases by A6,XXREAL_0:1;
          suppose
A30:        j1 = width G1;
            then p in { |[r,s]| : r <= G1*(1,1)`1 & G1*(1,width G1)`2 <= s }
            by A4,A16,A25,A27,A29;
            hence thesis by A17,A30,GOBRD11:25;
          end;
          suppose
A31:        j1 < width G1;
            then 1 <= j1 + 1 & j1 + 1 <= width G1 by NAT_1:11,13;
            then G1*(i1,j1+1) in Values G1 by A14,A15,MATRIX_0:41;
            then
            G2*(1,j2+1)`2 <= G1*(1,j1+1)`2 by A1,A4,A15,A7,A5,A9,A16,A24,A31
,Th8;
            then s9 <= G1*(1,j1+1)`2 by A28,XXREAL_0:2;
            then
            p in {|[r,s]|: r <= G1*(1,1)`1 & G1*(1,j1)`2 <= s & s <= G1*(
            1,j1+1)`2} by A4,A16,A25,A27,A29;
            hence thesis by A7,A17,A31,GOBRD11:26;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  suppose that
A32: i1 = 1 and
A33: 1 < i2;
A34: i1-'1 = 0 by A32,XREAL_1:232;
A35: 1 <= i2-'1 by A33,NAT_D:49;
    then i2-'1 < i2 by NAT_D:51;
    then
A36: i2-'1 < len G2 by A5,XXREAL_0:2;
A37: i2-'1+1 = i2 by A33,XREAL_1:235;
    now
      per cases by A8,XXREAL_0:1;
      suppose
A38:    j2 = width G2;
        then
        p in { |[r,s]|: G2*(i2-'1,1)`1 <= r & r <= G2*(i2,1)`1 & G2*(1,j2
        )`2 <= s} by A13,A35,A36,A37,GOBRD11:31;
        then consider r9,s9 such that
A39:    p = |[r9,s9]| and
        G2*(i2-'1,1)`1 <= r9 and
A40:    r9 <= G2*(i2,1)`1 & G2*(1,j2)`2 <= s9;
        r9 <= G1*(1,1)`1 & G1*(1,j1)`2 <= s9 by A4,A15,A7,A6,A11,A12,A32,A40,
GOBOARD5:2;
        then
A41:    p in { |[r,s]| : r <= G1*(1,1)`1 & G1*(1,j1)`2 <= s } by A39;
        j1 = width G1 by A1,A2,A4,A10,A5,A38,Th5;
        hence thesis by A34,A41,GOBRD11:25;
      end;
      suppose
A42:    j2 < width G2;
        then p in { |[r,s]| : G2*(i2-'1,1)`1 <= r & r <= G2*(i2,1)`1 & G2*(1,
        j2)`2 <= s & s <= G2*(1,j2+1)`2 } by A13,A9,A35,A36,A37,GOBRD11:32;
        then consider r9,s9 such that
A43:    p = |[r9,s9]| and
        G2*(i2-'1,1)`1 <= r9 and
A44:    r9 <= G2*(i2,1)`1 & G2*(1,j2)`2 <= s9 and
A45:    s9 <= G2*(1,j2+1)`2;
A46:    r9 <= G1*(1,1)`1 & G1*(1,j1)`2 <= s9 by A4,A15,A7,A6,A11,A12,A32,A44,
GOBOARD5:2;
        now
          per cases by A6,XXREAL_0:1;
          suppose
A47:        j1 = width G1;
            then p in { |[r,s]| : r <= G1*(1,1)`1 & G1*(1,width G1)`2 <= s }
            by A43,A46;
            hence thesis by A34,A47,GOBRD11:25;
          end;
          suppose
A48:        j1 < width G1;
            1 <= j2+1 & j2+1 <= width G2 by A42,NAT_1:12,13;
            then
A49:        G2*(i2,j2+1)`2 = G2*(1,j2+1)`2 by A10,A5,GOBOARD5:1;
            1 <= j1+1 & j1+1 <= width G1 by A48,NAT_1:12,13;
            then
            G1*(i1,j1+1) in Values G1 & G1*(i1,j1+1)`2 = G1*(1,j1+1)`2 by A14
,A15,GOBOARD5:1,MATRIX_0:41;
            then
            G2*(1,j2+1)`2 <= G1*(1,j1+1)`2 by A1,A4,A14,A15,A7,A10,A5,A9,A42
,A48,A49,Th8;
            then s9 <= G1*(1,j1+1)`2 by A45,XXREAL_0:2;
            then
            p in {|[r,s]|: r <= G1*(1,1)`1 & G1*(1,j1)`2 <= s & s <= G1*(
            1,j1+1)`2} by A43,A46;
            hence thesis by A7,A34,A48,GOBRD11:26;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  suppose
    1 < i1 & i2 = 1;
    hence thesis by A1,A2,A4,A9,A8,Th2;
  end;
  suppose
A50: 1 < i1 & 1 < i2;
    then
A51: 1 <= i2-'1 by NAT_D:49;
    then
A52: i2-'1+1 = i2 by NAT_D:43;
    i2-'1 < i2 by A51,NAT_D:51;
    then
A53: i2-'1 < len G2 by A5,XXREAL_0:2;
    then
A54: G2*(i2-'1,1)`1 = G2*(i2-'1,j2)`1 by A9,A8,A51,GOBOARD5:2;
A55: 1 <= i1-'1 by A50,NAT_D:49;
    then
A56: i1-'1+1 = i1 by NAT_D:43;
    i1-'1 < i1 by A55,NAT_D:51;
    then
A57: i1-'1 < len G1 by A15,XXREAL_0:2;
    then G1*(i1-'1,j1) in Values G1 & G1*(i1-'1,1)`1 = G1*(i1-'1,j1)`1 by A7,A6
,A55,GOBOARD5:2,MATRIX_0:41;
    then
A58: G1*(i1-'1,1)`1 <= G2*(i2-'1,1)`1 by A1,A4,A15,A7,A6,A5,A9,A8,A50,A54,Th7;
    now
      per cases by A8,XXREAL_0:1;
      suppose
A59:    j2 = width G2;
        then
        p in { |[r,s]|: G2*(i2-'1,1)`1 <= r & r <= G2*(i2,1)`1 & G2*(1,j2
        )`2 <= s} by A13,A51,A53,A52,GOBRD11:31;
        then consider r9,s9 such that
A60:    p = |[r9,s9]| and
A61:    G2*(i2-'1,1)`1 <= r9 & r9 <= G2*(i2,1)`1 and
A62:    G2*(1,j2)`2 <= s9;
A63:    G1*(1,j1)`2 <= s9 by A4,A14,A15,A7,A6,A12,A62,GOBOARD5:1;
        G1*(i1-'1,1)`1 <= r9 & r9 <= G1*(i1,1)`1 by A4,A14,A15,A7,A6,A11,A58
,A61,GOBOARD5:2,XXREAL_0:2;
        then
A64:    p in { |[r,s]|: G1*(i1-'1,1)`1 <= r & r <= G1*(i1,1)`1 & G1*(1,
        j1)`2 <= s} by A60,A63;
        j1 = width G1 by A1,A2,A4,A10,A5,A59,Th5;
        hence thesis by A55,A57,A56,A64,GOBRD11:31;
      end;
      suppose
A65:    j2 < width G2;
        then p in { |[r,s]| : G2*(i2-'1,1)`1 <= r & r <= G2*(i2,1)`1 & G2*(1,
        j2)`2 <= s & s <= G2*(1,j2+1)`2 } by A13,A9,A51,A53,A52,GOBRD11:32;
        then consider r9,s9 such that
A66:    p = |[r9,s9]| and
A67:    G2*(i2-'1,1)`1 <= r9 & r9 <= G2*(i2,1)`1 and
A68:    G2*(1,j2)`2 <= s9 and
A69:    s9 <= G2*(1,j2+1)`2;
A70:    G1*(1,j1)`2 <= s9 by A4,A14,A15,A7,A6,A12,A68,GOBOARD5:1;
A71:    G1*(i1-'1,1)`1 <= r9 & r9 <= G1*(i1,1)`1 by A4,A14,A15,A7,A6,A11,A58
,A67,GOBOARD5:2,XXREAL_0:2;
          per cases by A6,XXREAL_0:1;
          suppose
A72:        j1 = width G1;
            p in { |[r,s]|: G1*(i1-'1,1)`1 <= r & r <= G1*(i1,1)`1 & G1*
            (1,j1)`2 <= s } by A66,A71,A70;
            hence thesis by A55,A57,A56,A72,GOBRD11:31;
          end;
          suppose
A73:        j1 < width G1;
            1 <= j2+1 & j2+1 <= width G2 by A65,NAT_1:12,13;
            then
A74:        G2*(i2,j2+1)`2 = G2*(1,j2+1)`2 by A10,A5,GOBOARD5:1;
            1 <= j1+1 & j1+1 <= width G1 by A73,NAT_1:12,13;
            then G1*(i1,j1+1) in Values G1 & G1*(i1,j1+1)`2 = G1*(1,j1+1)`2
            by A14,A15,GOBOARD5:1,MATRIX_0:41;
            then G2*(1,j2+1)`2 <= G1*(1,j1+1)`2 by A1,A4,A14,A15,A7,A10,A5,A9
,A65,A73,A74,Th8;
            then s9 <= G1*(1,j1+1)`2 by A69,XXREAL_0:2;
            then p in { |[r,s]| : G1*(i1-'1,1)`1 <= r & r <= G1*(i1,1)`1 & G1
            *(1,j1)`2 <= s & s <= G1*(1,j1+1)`2 } by A66,A71,A70;
            hence thesis by A7,A55,A57,A56,A73,GOBRD11:32;
          end;
      end;
    end;
    hence thesis;
  end;
end;
