reserve i,j,k,n,m for Nat;
reserve p,q for Point of TOP-REAL 2;
reserve G for Go-board;
reserve C for Subset of TOP-REAL 2;

theorem
  for f being S-Sequence_in_R2, p being Point of TOP-REAL 2 st 1 <j & j
  <= len f & p in L~mid(f,1,j) holds LE p, f/.j, L~f, f/.1, f/.len f
proof
  let f be S-Sequence_in_R2, p be Point of TOP-REAL 2 such that
A1: 1 <j and
A2: j <= len f and
A3: p in L~mid(f,1,j);
  consider i such that
A4: 1 <= i and
A5: i+1 <= len mid(f,1,j) and
A6: p in LSeg(mid(f,1,j),i) by A3,SPPOL_2:13;
A7: j -' 1 + 1 = j by A1,XREAL_1:235;
  then
A8: i+1 <= j by A1,A2,A5,FINSEQ_6:186;
  then i < j-'1+1 by A7,NAT_1:13;
  then
A9: LSeg(mid(f,1,j),i) = LSeg(f,i+1-'1) by A1,A2,A4,JORDAN4:19;
  1 <= i+1 by NAT_1:11;
  then
A10: LE f/.(i+1), f/.j, L~f, f/.1, f/.len f by A2,A8,JORDAN5C:24;
A11: i = i + 1 -' 1 by NAT_D:34;
  i+1 <= len f by A2,A8,XXREAL_0:2;
  then LE p, f/.(i+1), L~f, f/.1, f/.len f by A4,A6,A11,A9,JORDAN5C:26;
  hence thesis by A10,JORDAN5C:13;
end;
