reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem
  for P1 being Subset of TOP-REAL n, P being Subset of TOP-REAL n,
  Q being Subset of (TOP-REAL n)|P, p1,p2 being Point of TOP-REAL n st
  P1 is_an_arc_of p1,p2 & P1 c=P & Q=P1\{p1,p2} holds Q is connected
proof
  let P1 be Subset of TOP-REAL n, P be Subset of TOP-REAL n,
  Q be Subset of (TOP-REAL n)|P, p1,p2 be Point of TOP-REAL n;
  assume that
A1: P1 is_an_arc_of p1,p2 and
A2: P1 c=P and
A3: Q=P1\{p1,p2};
  [#]((TOP-REAL n)|P1)=P1 by PRE_TOPC:def 5;
  then reconsider Q9=Q as Subset of (TOP-REAL n)|P1 by A3,XBOOLE_1:36;
  Q9 is connected by A1,A3,Th40;
  hence thesis by A2,Th41;
end;
