reserve n for Nat;

theorem Th50:
  for f be S-Sequence_in_R2 for Q be closed Subset of TOP-REAL 2
  st L~f meets Q & not f/.1 in Q & First_Point(L~f,f/.1,f/.len f,Q) in rng f
holds L~mid(f,1,First_Point(L~f,f/.1,f/.len f,Q)..f) /\ Q = {First_Point(L~f,f
  /.1,f/.len f,Q)}
proof
  let f be S-Sequence_in_R2;
  let Q be closed Subset of TOP-REAL 2;
  assume that
A1: L~f meets Q & not f/.1 in Q and
A2: First_Point(L~f,f/.1,f/.len f,Q) in rng f;
  L~R_Cut(f,First_Point(L~f,f/.1,f/.len f,Q)) /\ Q = {First_Point(L~f,f/.1
  ,f/.len f,Q)} by A1,SPRECT_4:1;
  hence thesis by A2,Th49;
end;
