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 Th41:
  for GX being TopSpace, P1, P being Subset of GX, Q9 being Subset of GX|P1,
  Q being Subset of GX|P st P1 c=P & Q=Q9 &
  Q9 is connected holds Q is connected
proof
  let GX be TopSpace, P1, P be Subset of GX, Q9 be Subset of GX|P1,
  Q be Subset of GX|P;
  assume that
A1: P1 c=P and
A2: Q=Q9 and
A3: Q9 is connected;
  [#](GX|P)=P by PRE_TOPC:def 5;
  then reconsider P19=P1 as Subset of GX|P by A1;
  GX|P1=(GX|P)|P19 by A1,PRE_TOPC:7;
  hence thesis by A2,A3,CONNSP_1:23;
end;
