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

theorem Th25:
  for f being special non empty FinSequence of TOP-REAL 2 holds (m
  ,n)-cut f is special
proof
  let f being special non empty FinSequence of TOP-REAL 2;
  set h = (m,n)-cut f;
  let i be Nat such that
A1: 1 <= i and
A2: i+1 <= len h;
  per cases;
  suppose
    not(1<=m & m<=n & n<=len f);
    then h = {} by FINSEQ_6:def 4;
    hence thesis by A2;
  end;
  suppose
A3: 1<=m & m<=n & n<=len f;
    then
A4: 1+1 <= i+m by A1,XREAL_1:7;
    then
A5: 1 <= i+m by XXREAL_0:2;
A6: i-'1+m = i+m-'1 by A1,NAT_D:38;
    then
A7: 1 <= i-'1+m by A4,NAT_D:55;
A8: i-'1+m+1 = i+m-'1+1 by A1,NAT_D:38
      .= i+m by A4,XREAL_1:235,XXREAL_0:2;
A9: i < len h by A2,NAT_1:13;
    len h +m = n+1 by A3,FINSEQ_6:def 4;
    then i+m < n+1 by A9,XREAL_1:6;
    then i+m <= n by NAT_1:13;
    then
A10: i+m <= len f by A3,XXREAL_0:2;
    then
A11: i-'1+m <= len f by A6,NAT_D:44;
    i -' 1 <= i by NAT_D:44;
    then
A12: i -' 1 < len h by A9,XXREAL_0:2;
A13: h/.(i+1) =h.(i+1) by A2,FINSEQ_4:15,NAT_1:11
      .= f.(i+m) by A3,A9,FINSEQ_6:def 4
      .= f/.(i+m) by A5,A10,FINSEQ_4:15;
    i-'1+1 = i by A1,XREAL_1:235;
    then h/.i =h.(i -' 1+1) by A1,A9,FINSEQ_4:15
      .= f.(i-'1+m) by A3,A12,FINSEQ_6:def 4
      .= f/.(i-'1+m) by A6,A4,A11,FINSEQ_4:15,NAT_D:55;
    hence thesis by A7,A10,A13,A8,TOPREAL1:def 5;
  end;
end;
