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
  (m in dom f implies 1 <= len(f|m)) &
  (f is_sequence_on M implies f|m is_sequence_on M)
proof
  set g=f|m;
  thus m in dom f implies 1 <= len(f|m)
  proof
    assume m in dom f;
    then 1<=m & m<=len f by FINSEQ_3:25;
    hence thesis by FINSEQ_1:59;
  end;
  assume
A1: f is_sequence_on M;
  per cases;
  suppose
A2: m < 1;
    m = 0 by A2,NAT_1:14;
    hence thesis;
  end;
  suppose
    m >= len f;
    hence thesis by A1,FINSEQ_1:58;
  end;
  suppose
A3: 1 <= m & m < len f;
A4: dom g = Seg len g by FINSEQ_1:def 3;
A5: m in dom f & len g = m by A3,FINSEQ_1:59,FINSEQ_3:25;
A6: now
      let n;
      assume
A7:   n in dom g & n+1 in dom g;
      then
A8:   n in dom f & n+1 in dom f by A4,A5,FINSEQ_4:71;
      let i1,i2,j1,j2 be Nat;
      assume
A9:  [i1,i2] in Indices M & [j1,j2] in Indices M & g/.n=M*(i1,i2) &
      g/.(n+ 1)=M*(j1,j2);
      g/.n=f/.n & g/.(n+1)=f/.(n+1) by A4,A5,A7,FINSEQ_4:71;
      hence |.i1-j1.|+|.i2-j2.| = 1 by A1,A8,A9;
    end;
    now
      let n;
      assume
A10:  n in dom g;
      then n in dom f by A4,A5,FINSEQ_4:71;
      then consider i,j such that
A11:  [i,j] in Indices M & f/.n=M*(i,j) by A1;
      take i,j;
      thus [i,j] in Indices M & g/.n = M*(i,j) by A4,A5,A10,A11,FINSEQ_4:71;
    end;
    hence thesis by A6;
  end;
end;
