reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;

theorem Th11:
  for P being Subset of TOP-REAL n, p1,p2,q1 being Point of
TOP-REAL n st P is_an_arc_of p2,p1 & LSeg(q1,p2) /\ P = {p2} holds LSeg(q1,p2)
  \/ P is_an_arc_of q1,p1
proof
  let P be Subset of TOP-REAL n, p1,p2,q1 be Point of TOP-REAL n;
  assume that
A1: P is_an_arc_of p2,p1 and
A2: LSeg(q1,p2) /\ P = {p2};
  per cases;
  suppose
    p2 <> q1;
    then LSeg(q1,p2) is_an_arc_of q1,p2 by Th9;
    hence thesis by A1,A2,Th2;
  end;
  suppose
A3: p2 = q1;
    then
A4: LSeg(q1,p2) = {q1} by RLTOPSP1:70;
    q1 in P by A1,A3,Th1;
    hence thesis by A1,A3,A4,ZFMISC_1:40;
  end;
end;
