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 Th12:
  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,
  j2-'1) c= cell(G1,i1,j1-'1)
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: 1 <= i1 by A2,MATRIX_0:32;
A6: 1 <= j2 by A3,MATRIX_0:32;
A7: 1 <= i2 by A3,MATRIX_0:32;
A8: j1 <= width G1 by A2,MATRIX_0:32;
A9: j2 <= width G2 by A3,MATRIX_0:32;
A10: i2 <= len G2 by A3,MATRIX_0:32;
  then
A11: G2*(i2,j2)`1 = G2*(i2,1)`1 by A7,A6,A9,GOBOARD5:2;
A12: i1 <= len G1 by A2,MATRIX_0:32;
A13: 1 <= j1 by A2,MATRIX_0:32;
then A14: G1*(i1,j1)`2 = G1*(1,j1)`2 by A5,A12,A8,GOBOARD5:1;
  let p be object such that
A15: p in cell(G2,i2,j2-'1);
A16: G2*(i2,j2)`2 = G2*(1,j2)`2 by A7,A10,A6,A9,GOBOARD5:1;
  per cases by A13,A6,XXREAL_0:1;
  suppose
A17: j1 = 1 & j2 = 1;
    then
A18: j1-'1 = 0 by XREAL_1:232;
A19: j2-'1 = 0 by A17,XREAL_1:232;
    now
      per cases by A10,XXREAL_0:1;
      suppose
A20:    i2 = len G2;
        then p in { |[r,s]| : G2*(len G2,1)`1 <= r & s <= G2*(1,1)`2 } by A15
,A19,GOBRD11:27;
        then consider r9,s9 such that
A21:    p = |[r9,s9]| & G2*(len G2,1)`1 <= r9 and
A22:    s9 <= G2*(1,1)`2;
A23:    i1 = len G1 by A1,A2,A4,A6,A9,A20,Th3;
        G2*(1,1)`2 = G2*(i2,j2)`2 by A7,A10,A9,A17,GOBOARD5:1;
        then s9 <= G1*(1,1)`2 by A4,A5,A12,A8,A17,A22,GOBOARD5:1;
        then p in { |[r,s]| : G1*(len G1,1)`1 <= r & s <= G1*(1,1)`2 } by A4
,A17,A20,A21,A23;
        hence thesis by A18,A23,GOBRD11:27;
      end;
      suppose
A24:    i2 < len G2;
        then
        p in {|[r,s]|: G2*(i2,1)`1 <= r & r <= G2*(i2+1,1)`1 & s <= G2*(1
        ,1)`2 } by A15,A7,A19,GOBRD11:30;
        then consider r9,s9 such that
A25:    p = |[r9,s9]| & G2*(i2,1)`1 <= r9 and
A26:    r9 <= G2*(i2+1,1)`1 and
A27:    s9 <= G2*(1,1)`2;
        G2*(1,1)`2 = G2*(i2,j1)`2 by A7,A10,A9,A17,GOBOARD5:1;
        then
A28:    s9 <= G1*(1,1)`2 by A4,A5,A12,A8,A17,A27,GOBOARD5:1;
        now
          per cases by A12,XXREAL_0:1;
          suppose
A29:        i1 = len G1;
            then p in { |[r,s]| : G1*(len G1,1)`1 <= r & s <= G1*(1,1)`2 } by
A4,A17,A25,A28;
            hence thesis by A18,A29,GOBRD11:27;
          end;
          suppose
A30:        i1 < len G1;
            then
            G2*(i2+1,1)`1 <= G1*(i1+1,1)`1 by A1,A4,A5,A8,A7,A9,A17,A24,Th6;
            then r9 <= G1*(i1+1,1)`1 by A26,XXREAL_0:2;
            then p in {|[r,s]|: G1*(i1,1)`1 <= r & r <= G1*(i1+1,1)`1 & s <=
            G1*(1,1)`2} by A4,A17,A25,A28;
            hence thesis by A5,A18,A30,GOBRD11:30;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  suppose that
A31: j1 = 1 and
A32: 1 < j2;
A33: j1-'1 = 0 by A31,XREAL_1:232;
A34: 1 <= j2-'1 by A32,NAT_D:49;
    then j2-'1 < j2 by NAT_D:51;
    then
A35: j2-'1 < width G2 by A9,XXREAL_0:2;
A36: j2-'1+1 = j2 by A32,XREAL_1:235;
    now
      per cases by A10,XXREAL_0:1;
      suppose
A37:    i2 = len G2;
        then
        p in { |[r,s]| : G2*(i2,1)`1 <= r & G2*(1,j2-'1)`2 <= s & s <= G2
        *(1,j2)`2 } by A15,A34,A35,A36,GOBRD11:29;
        then
        ex r9,s9 st p = |[r9,s9]| & G2*(i2,1)`1 <= r9 & G2*(1,j2-'1)`2 <=
        s9 & s9 <= G2*(1,j2)`2;
        then
A38:    p in { |[r,s]| : G1*(i1,1)`1 <= r & s <= G1*(1,1)`2 } by A4,A14,A11,A16
,A31;
        i1 = len G1 by A1,A2,A4,A6,A9,A37,Th3;
        hence thesis by A33,A38,GOBRD11:27;
      end;
      suppose
A39:    i2 < len G2;
        then
        p in { |[r,s]| : G2*(i2,1)`1 <= r & r <= G2*(i2+1,1)`1 & G2*(1,j2
        -'1)`2 <= s & s <= G2*(1,j2)`2 } by A15,A7,A34,A35,A36,GOBRD11:32;
        then consider r9,s9 such that
A40:    p = |[r9,s9]| and
A41:    G2*(i2,1)`1 <= r9 and
A42:    r9 <= G2*(i2+1,1)`1 and
        G2*(1,j2-'1)`2 <= s9 and
A43:    s9 <= G2*(1,j2)`2;
A44:    s9 <= G1*(1,1)`2 & G1*(i1,1)`1 <= r9 by A4,A7,A10,A6,A9,A14,A31,A41,A43
,GOBOARD5:1,2;
        now
          per cases by A12,XXREAL_0:1;
          suppose
A45:        i1 = len G1;
            then p in { |[r,s]| : G1*(len G1,1)`1 <= r & s <= G1*(1,1)`2 } by
A40,A44;
            hence thesis by A33,A45,GOBRD11:27;
          end;
          suppose
A46:        i1 < len G1;
            1 <= i2+1 & i2+1 <= len G2 by A39,NAT_1:12,13;
            then
A47:        G2*(i2+1,j2)`1 = G2*(i2+1,1)`1 by A6,A9,GOBOARD5:2;
            1 <= i1+1 & i1+1 <= len G1 by A46,NAT_1:12,13;
            then G1*(i1+1,j1)`1 = G1*(i1+1,1)`1 by A13,A8,GOBOARD5:2;
            then G2*(i2+1,1)`1 <= G1*(i1+1,1)`1 by A1,A4,A5,A13,A8,A7,A6,A9,A39
,A46,A47,Th6;
            then r9 <= G1*(i1+1,1)`1 by A42,XXREAL_0:2;
            then p in {|[r,s]|: G1*(i1,1)`1 <= r & r <= G1*(i1+1,1)`1 & s <=
            G1*(1,1)`2} by A40,A44;
            hence thesis by A5,A33,A46,GOBRD11:30;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  suppose
    1 < j1 & j2 = 1;
    hence thesis by A1,A2,A4,A7,A10,Th4;
  end;
  suppose
A48: 1 < j1 & 1 < j2;
    then
A49: 1 <= j2-'1 by NAT_D:49;
    then
A50: j2-'1+1 = j2 by NAT_D:43;
    j2-'1 < j2 by A49,NAT_D:51;
    then
A51: j2-'1 < width G2 by A9,XXREAL_0:2;
    then
A52: G2*(1,j2-'1)`2 = G2*(i2,j2-'1)`2 by A7,A10,A49,GOBOARD5:1;
A53: 1 <= j1-'1 by A48,NAT_D:49;
    then
A54: j1-'1+1 = j1 by NAT_D:43;
    j1-'1 < j1 by A53,NAT_D:51;
    then
A55: j1-'1 < width G1 by A8,XXREAL_0:2;
    then G1*(1,j1-'1)`2 = G1*(i1,j1-'1)`2 by A5,A12,A53,GOBOARD5:1;
    then
A56: G1*(1,j1-'1)`2 <= G2*(1,j2-'1)`2 by A1,A4,A5,A12,A8,A7,A10,A9,A48,A52,Th9;
    now
      per cases by A10,XXREAL_0:1;
      suppose
A57:    i2 = len G2;
        then
        p in { |[r,s]| : G2*(i2,1)`1 <= r & G2*(1,j2-'1)`2 <= s & s <= G2
        * (1,j2)`2} by A15,A49,A51,A50,GOBRD11:29;
        then consider r9,s9 such that
A58:    p = |[r9,s9]| and
A59:    G2*(i2,1)`1 <= r9 and
A60:    G2*(1,j2-'1)`2 <= s9 & s9 <= G2*(1,j2)`2;
A61:    G1*(i1,1)`1 <= r9 by A4,A5,A12,A13,A8,A11,A59,GOBOARD5:2;
        G1*(1,j1-'1)`2 <= s9 & s9 <= G1*(1,j1)`2 by A4,A5,A12,A13,A8,A16,A56
,A60,GOBOARD5:1,XXREAL_0:2;
        then
A62:    p in {|[r,s]|: G1*(i1,1)`1 <= r & G1*(1,j1-'1)`2 <= s & s <= G1*(
        1,j1)`2} by A58,A61;
        i1 = len G1 by A1,A2,A4,A6,A9,A57,Th3;
        hence thesis by A53,A55,A54,A62,GOBRD11:29;
      end;
      suppose
A63:    i2 < len G2;
        then
        p in { |[r,s]| : G2*(i2,1)`1 <= r & r <= G2*(i2+1,1)`1 & G2*(1,j2
        -'1)`2 <= s & s <= G2*(1,j2)`2 } by A15,A7,A49,A51,A50,GOBRD11:32;
        then consider r9,s9 such that
A64:    p = |[r9,s9]| and
A65:    G2*(i2,1)`1 <= r9 and
A66:    r9 <= G2*(i2+1,1)`1 and
A67:    G2*(1,j2-'1)`2 <= s9 & s9 <= G2*(1,j2)`2;
A68:    G1*(i1,1)`1 <= r9 by A4,A5,A12,A13,A8,A11,A65,GOBOARD5:2;
A69:    G1*(1,j1-'1)`2 <= s9 & s9 <= G1*(1,j1)`2 by A4,A5,A12,A13,A8,A16,A56
,A67,GOBOARD5:1,XXREAL_0:2;
        now
          per cases by A12,XXREAL_0:1;
          suppose
A70:        i1 = len G1;
            p in {|[r,s]|: G1*(i1,1)`1 <= r & G1*(1,j1-'1)`2 <= s & s <=
            G1* (1,j1)`2} by A64,A69,A68;
            hence thesis by A53,A55,A54,A70,GOBRD11:29;
          end;
          suppose
A71:        i1 < len G1;
            1 <= i2+1 & i2+1 <= len G2 by A63,NAT_1:12,13;
            then
A72:        G2*(i2+1,j2)`1 = G2*(i2+1,1)`1 by A6,A9,GOBOARD5:2;
            1 <= i1+1 & i1+1 <= len G1 by A71,NAT_1:12,13;
            then G1*(i1+1,j1)`1 = G1*(i1+1,1)`1 by A13,A8,GOBOARD5:2;
            then G2*(i2+1,1)`1 <= G1*(i1+1,1)`1 by A1,A4,A5,A13,A8,A7,A6,A9,A63
,A71,A72,Th6;
            then r9 <= G1*(i1+1,1)`1 by A66,XXREAL_0:2;
            then p in { |[r,s]| : G1*(i1,1)`1 <= r & r <= G1*(i1+1,1)`1 & G1*
            (1,j1-'1)`2 <= s & s <= G1*(1,j1)`2 } by A64,A69,A68;
            hence thesis by A5,A53,A55,A54,A71,GOBRD11:32;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
end;
