reserve p for Point of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  v, v1,v2 for FinSequence of REAL,
  r,s for Real,
  n,m,i,j,k for Nat,
  x for set;
reserve G for Go-board;
reserve D for set,
  f for FinSequence of D,
  M for Matrix of D;

theorem
  (for n st n in dom f1 ex i,j st [i,j] in Indices M & f1/.n=M*(i,j)) &
(for n st n in dom f2 ex i,j st [i,j] in Indices M & f2/.n=M*(i,j)) implies for
  n st n in dom(f1^f2) ex i,j st [i,j] in Indices M & (f1^f2)/.n=M*(i,j)
proof
  assume that
A1: for n st n in dom f1 ex i,j st [i,j] in Indices M & f1/.n=M*(i,j) and
A2: for n st n in dom f2 ex i,j st [i,j] in Indices M & f2/.n=M*(i,j);
  let n such that
A3: n in dom(f1^f2);
  per cases by A3,FINSEQ_1:25;
  suppose
A4: n in dom f1;
    then consider i,j such that
A5: [i,j] in Indices M and
A6: f1/.n=M*(i,j) by A1;
    take i,j;
    thus [i,j] in Indices M by A5;
    thus thesis by A4,A6,FINSEQ_4:68;
  end;
  suppose
    ex m be Nat st m in dom f2 & n=len f1+m;
    then consider m be Nat such that
A7: m in dom f2 and
A8: n=len f1+m;
    consider i,j such that
A9: [i,j] in Indices M and
A10: f2/.m=M*(i,j) by A2,A7;
    take i,j;
    thus [i,j] in Indices M by A9;
    thus thesis by A7,A8,A10,FINSEQ_4:69;
  end;
end;
