reserve i,j,l for Nat;

theorem Th3:
  for f being S-Sequence_in_R2, p being Point of TOP-REAL 2, j st 1
  <=j & j < len f & p in L~mid(f,j,len f) holds LE f/.j, p, L~f, f/.1, f/.len f
proof
  let f be S-Sequence_in_R2, p be Point of TOP-REAL 2, j such that
A1: 1 <=j and
A2: j < len f and
A3: p in L~mid(f,j,len f);
  consider i such that
A4: 1 <= i and
A5: i+1 <= len mid(f,j,len f) and
A6: p in LSeg(mid(f,j,len f),i) by A3,SPPOL_2:13;
A7: len mid(f,j,len f)=len f-'j+1 by A1,A2,FINSEQ_6:186;
  then i <= len f -' j by A5,XREAL_1:6;
  then
A8: i + j <= len f by A2,NAT_D:54;
  j+i >= i by NAT_1:11;
  then
A9: j+i-'1+1 <= len f by A4,A8,XREAL_1:235,XXREAL_0:2;
  1+1 <= j+i by A1,A4,XREAL_1:7;
  then
A10: 1 <= j+i-'1 by NAT_D:49;
  j+1 <= j+i by A4,XREAL_1:6;
  then
A11: j <= j+i-'1 by NAT_D:49;
  j+i-'1+1 >= j+i-'1 by NAT_1:11;
  then len f >= j+i-'1 by A9,XXREAL_0:2;
  then
A12: LE f/.j, f/.(j+i-'1), L~f, f/.1, f/.len f by A1,A11,JORDAN5C:24;
  i<len f-'j+1 by A5,A7,NAT_1:13;
  then p in LSeg(f,j+i-'1) by A1,A2,A4,A6,JORDAN4:19;
  then LE f/.(j+i-'1), p, L~f, f/.1, f/.len f by A10,A9,JORDAN5C:25;
  hence thesis by A12,JORDAN5C:13;
end;
