reserve G for Go-board,
  i,j,k,m,n for Nat;

theorem Th13:
  for D being set, f being FinSequence of D st 1 <= k holds
  (k+1,len f)-cut f = f/^k
proof
  let D be set, f being FinSequence of D such that
A1: 1 <= k;
  per cases;
  suppose
A2: len f < k;
    k <= k+1 by NAT_1:11;
    then len f < k+1 by A2,XXREAL_0:2;
    hence (k+1, len f)-cut f = {} by FINSEQ_6:def 4
      .= f/^k by A2,FINSEQ_5:32;
  end;
  suppose
A3: f = {};
    then
A4: len f = 0;
    thus (k+1, len f)-cut f = <*>D by A3,FINSEQ_6:def 4
      .= f/^k by A1,A4,RFINSEQ:def 1;
  end;
  suppose that
A5: k <= len f;
    set IT = (k+1, len f)-cut f;
A6: 1 <= k+1 by NAT_1:11;
A7: k+1 <= len f + 1 by A5,XREAL_1:6;
A8: for m be Nat st m in dom IT holds IT.m = f.(m+k)
    proof
      let m be Nat such that
A9:   m in dom IT;
      1 <= m by A9,FINSEQ_3:25;
      then consider i be Nat such that
A10:  m = 1+i by NAT_1:10;
      reconsider i as Nat;
      m <= len IT by A9,FINSEQ_3:25;
      then i < len IT by A10,NAT_1:13;
      hence IT.m = f.(k+1+i) by A6,A7,A10,Lm1
        .= f.(m+k) by A10;
    end;
    len f + 1 = len IT + (k+1) by A6,A7,Lm1
      .= len IT + k+1;
    then len IT = len f - k;
    hence thesis by A5,A8,RFINSEQ:def 1;
  end;
end;
