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 Th10:
  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) c= cell(G1,i1,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: 1 <= i1 by A2,MATRIX_0:32;
A6: j1 <= width G1 by A2,MATRIX_0:32;
  let p be object such that
A7: p in cell(G2,i2,j2);
A8: 1 <= i2 by A3,MATRIX_0:32;
A9: j2 <= width G2 by A3,MATRIX_0:32;
A10: 1 <= j2 by A3,MATRIX_0:32;
A11: i2 <= len G2 by A3,MATRIX_0:32;
  then
A12: G2*(i2,j2)`1 = G2*(i2,1)`1 & G2*(i2,j2)`2 = G2*(1,j2)`2 by A8,A10,A9,
GOBOARD5:1,2;
A13: 1 <= j1 by A2,MATRIX_0:32;
A14: i1 <= len G1 by A2,MATRIX_0:32;
  then
A15: G1*(i1,j1)`1 = G1*(i1,1)`1 & G1*(i1,j1)`2 = G1*(1,j1)`2 by A5,A13,A6,
GOBOARD5:1,2;
  per cases by A11,A9,XXREAL_0:1;
  suppose
A16: i2 = len G2 & j2 = width G2;
    then
A17: p in { |[r,s]| : G2*(i2,j2)`1 <= r & G2*(i2,j2)`2 <= s } by A7,A12,
GOBRD11:28;
    i1 = len G1 & j1 = width G1 by A1,A2,A4,A8,A10,A16,Th3,Th5;
    hence thesis by A4,A15,A17,GOBRD11:28;
  end;
  suppose
A18: i2 = len G2 & j2 < width G2;
    then p in { |[r,s]| : G2*(i2,j2)`1 <= r & G2*(i2,j2)`2 <= s & s <= G2*(1,
    j2+1)`2 } by A7,A10,A12,GOBRD11:29;
    then consider r9,s9 such that
A19: p = |[r9,s9]| & G2*(i2,j2)`1 <= r9 & G2*(i2,j2)`2 <= s9 and
A20: s9 <= G2*(1,j2+1)`2;
A21: i1 = len G1 by A1,A2,A4,A10,A18,Th3;
    now
      per cases by A6,XXREAL_0:1;
      suppose
A22:    j1 = width G1;
        p in { |[r,s]| : G1*(i1,j1)`1 <= r & G1*(i1,j1)`2 <= s} by A4,A19;
        hence thesis by A15,A21,A22,GOBRD11:28;
      end;
      suppose
A23:    j1 < width G1;
        1 <= j2+1 & j2+1 <= width G2 by A18,NAT_1:12,13;
        then
A24:    G2*(i2,j2+1)`2 = G2*(1,j2+1)`2 by A8,A11,GOBOARD5:1;
        1 <= j1+1 & j1+1 <= width G1 by A23,NAT_1:12,13;
        then
        G1*(i1,j1+1) in Values G1 & G1*(i1,j1+1)`2 = G1*(1,j1+1)`2 by A5,A14,
GOBOARD5:1,MATRIX_0:41;
        then G2*(1,j2+1)`2 <= G1*(1,j1+1)`2 by A1,A4,A5,A14,A13,A8,A10,A18,A23
,A24,Th8;
        then s9 <= G1*(1,j1+1)`2 by A20,XXREAL_0:2;
        then
        p in {|[r,s]| : G1*(i1,j1)`1 <= r & G1*(i1,j1)`2 <= s & s <= G1*(
        1,j1+1)`2} by A4,A19;
        hence thesis by A13,A15,A21,A23,GOBRD11:29;
      end;
    end;
    hence thesis;
  end;
  suppose
A25: i2 < len G2 & j2 = width G2;
    then p in {|[r,s]|: G2*(i2,j2)`1 <= r & r <= G2*(i2+1,1)`1 & G2*(i2,j2)`2
    <= s} by A7,A8,A12,GOBRD11:31;
    then consider r9,s9 such that
A26: p = |[r9,s9]| & G2*(i2,j2)`1 <= r9 and
A27: r9 <= G2*(i2+1,1)`1 and
A28: G2*(i2,j2)`2 <= s9;
A29: j1 = width G1 by A1,A2,A4,A8,A25,Th5;
    now
      per cases by A14,XXREAL_0:1;
      suppose
A30:    i1 = len G1;
        p in { |[r,s]| : G1*(i1,j1)`1 <= r & G1*(i1,j1)`2 <= s} by A4,A26,A28;
        hence thesis by A15,A29,A30,GOBRD11:28;
      end;
      suppose
A31:    i1 < len G1;
        1 <= i2+1 & i2+1 <= len G2 by A25,NAT_1:12,13;
        then
A32:    G2*(i2+1,j2)`1 = G2*(i2+1,1)`1 by A10,A9,GOBOARD5:2;
        1 <= i1+1 & i1+1 <= len G1 by A31,NAT_1:12,13;
        then G1*(i1+1,j1)`1 = G1*(i1+1,1)`1 by A13,A6,GOBOARD5:2;
        then G2*(i2+1,1)`1 <= G1*(i1+1,1)`1 by A1,A4,A5,A13,A6,A8,A10,A25,A31
,A32,Th6;
        then r9 <= G1*(i1+1,1)`1 by A27,XXREAL_0:2;
        then
        p in {|[r,s]|: G1*(i1,j1)`1 <= r & r <= G1*(i1+1,1)`1 & G1*(i1,j1
        )`2 <= s} by A4,A26,A28;
        hence thesis by A5,A15,A29,A31,GOBRD11:31;
      end;
    end;
    hence thesis;
  end;
  suppose
A33: i2 < len G2 & j2 < width G2;
    then 1 <= j2+1 & j2+1 <= width G2 by NAT_1:12,13;
    then
A34: G2*(i2,j2+1)`2 = G2*(1,j2+1)`2 by A8,A11,GOBOARD5:1;
    1 <= i2+1 & i2+1 <= len G2 by A33,NAT_1:12,13;
    then G2*(i2+1,j2)`1 = G2*(i2+1,1)`1 by A10,A9,GOBOARD5:2;
    then
    p in { |[r,s]| : G2*(i2,j2)`1 <= r & r <= G2*(i2+1,j2)`1 & G2*(i2,j2)
    `2 <= s & s <= G2*(i2,j2+1)`2 } by A7,A8,A10,A12,A33,A34,GOBRD11:32;
    then consider r9,s9 such that
A35: p = |[r9,s9]| & G2*(i2,j2)`1 <= r9 and
A36: r9 <= G2*(i2+1,j2)`1 and
A37: G2*(i2,j2)`2 <= s9 and
A38: s9 <= G2*(i2,j2+1)`2;
    now
      per cases by A14,A6,XXREAL_0:1;
      suppose
A39:    i1 = len G1 & j1 = width G1;
        p in { |[r,s]| : G1*(i1,j1)`1 <= r & G1*(i1,j1)`2 <= s } by A4,A35,A37;
        hence thesis by A15,A39,GOBRD11:28;
      end;
      suppose
A40:    i1 = len G1 & j1 < width G1;
        then 1 <= j1+1 & j1+1 <= width G1 by NAT_1:12,13;
        then
        G1*(i1,j1+1) in Values G1 & G1*(i1,j1+1)`2 = G1*(1,j1+1)`2 by A5,A14,
GOBOARD5:1,MATRIX_0:41;
        then G2*(i2,j2+1)`2 <= G1*(1,j1+1)`2 by A1,A4,A5,A13,A8,A10,A33,A40
,Th8;
        then s9 <= G1*(1,j1+1)`2 by A38,XXREAL_0:2;
        then p in {|[r,s]| : G1*(i1,j1)`1 <= r & G1*(i1,j1)`2 <= s & s <= G1*
        (1,j1+1)`2} by A4,A35,A37;
        hence thesis by A13,A15,A40,GOBRD11:29;
      end;
      suppose
A41:    i1 < len G1 & j1 = width G1;
        then 1 <= i1+1 & i1+1 <= len G1 by NAT_1:12,13;
        then G1*(i1+1,j1)`1 = G1*(i1+1,1)`1 by A13,A6,GOBOARD5:2;
        then G2*(i2+1,j2)`1 <= G1*(i1+1,1)`1 by A1,A4,A5,A13,A8,A10,A33,A41
,Th6;
        then r9 <= G1*(i1+1,1)`1 by A36,XXREAL_0:2;
        then
        p in {|[r,s]|: G1*(i1,j1)`1 <= r & r <= G1*(i1+1,1)`1 & G1*(i1,j1
        )`2 <= s} by A4,A35,A37;
        hence thesis by A5,A15,A41,GOBRD11:31;
      end;
      suppose
A42:    i1 < len G1 & j1 < width G1;
        then 1 <= i1+1 & i1+1 <= len G1 by NAT_1:12,13;
        then G1*(i1+1,j1)`1 = G1*(i1+1,1)`1 by A13,A6,GOBOARD5:2;
        then G2*(i2+1,j2)`1 <= G1*(i1+1,1)`1 by A1,A4,A5,A13,A8,A10,A33,A42
,Th6;
        then
A43:    r9 <= G1*(i1+1,1)`1 by A36,XXREAL_0:2;
        1 <= j1+1 & j1+1 <= width G1 by A42,NAT_1:12,13;
        then
        G1*(i1,j1+1) in Values G1 & G1*(i1,j1+1)`2 = G1*(1,j1+1)`2 by A5,A14,
GOBOARD5:1,MATRIX_0:41;
        then G2*(i2,j2+1)`2 <= G1*(1,j1+1)`2 by A1,A4,A5,A13,A8,A10,A33,A42
,Th8;
        then s9 <= G1*(1,j1+1)`2 by A38,XXREAL_0:2;
        then
        p in { |[r,s]| : G1*(i1,1)`1 <= r & r <= G1*(i1+1,1)`1 & G1*(1,j1
        )`2 <= s & s <= G1*(1,j1+1)`2 } by A4,A15,A35,A37,A43;
        hence thesis by A5,A13,A42,GOBRD11:32;
      end;
    end;
    hence thesis;
  end;
end;
