reserve P,P1,P2,R for Subset of TOP-REAL 2,
  p,p1,p2,p3,p11,p22,q,q1,q2,q3,q4 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r for Real,
  u for Point of Euclid 2,
  n,m,i,j,k for Nat,
  x,y for set;
reserve P, R for Subset of TOP-REAL 2;

theorem Th24:
  for p st R is being_Region & P = {q: q<>p & q in R & not ex P1
  being Subset of TOP-REAL 2 st P1 is_S-P_arc_joining p,q & P1 c=R} holds P is
  open
proof
  let p;
  assume that
A1: R is being_Region and
A2: P = {q: q<>p & q in R & not ex P1 being Subset of TOP-REAL 2 st P1
  is_S-P_arc_joining p,q & P1 c=R};
  reconsider RR = R, PP=P as Subset of the TopStruct of TOP-REAL 2;
  R is open by A1;
  then
A3: RR is open by PRE_TOPC:30;
  now
    let u;
    reconsider p2=u as Point of TOP-REAL 2 by TOPREAL3:8;
    assume
A4: u in P;
    then
    ex q1 st q1 = u & q1<>p & q1 in R & not ex P1 being Subset of TOP-REAL
    2 st P1 is_S-P_arc_joining p,q1 & P1 c=R by A2;
    then consider r being Real such that
A5: r>0 and
A6: Ball(u,r) c= RR by A3,Lm1,TOPMETR:15;
    take r;
    thus r>0 by A5;
    reconsider r9 = r as Real;
A7: p2 in Ball(u,r9) by A5,TBSP_1:11;
    thus Ball(u,r) c= P
    proof
      assume not thesis;
      then consider x being object such that
A8:   x in Ball(u,r) and
A9:   not x in P;
      x in R by A6,A8;
      then reconsider q=x as Point of TOP-REAL 2;
      now
        per cases by A2,A6,A8,A9;
        suppose
A10:      q=p;
A11:      now
            assume
A12:        q=p2;
            ex p3 st p3=p2 & p3<>p & p3 in R & not ex P1 being Subset of
            TOP-REAL 2 st P1 is_S-P_arc_joining p,p3 & P1 c=R by A2,A4;
            hence contradiction by A10,A12;
          end;
          u in Ball(u,r9) by A5,TBSP_1:11;
          then
A13:      ex P2 being Subset of TOP-REAL 2 st P2 is_S-P_arc_joining q,p2 &
          P2 c= Ball(u,r9) by A8,A11,Th10;
          not p2 in P
          proof
            assume p2 in P;
            then ex q4 st q4=p2 & q4<>p & q4 in R & not ex P1 being Subset of
            TOP-REAL 2 st P1 is_S-P_arc_joining p,q4 & P1 c=R by A2;
            hence contradiction by A6,A10,A13,XBOOLE_1:1;
          end;
          hence contradiction by A4;
        end;
        suppose
A14:      ex P1 being Subset of TOP-REAL 2 st P1 is_S-P_arc_joining p
          ,q & P1 c=R;
          not p2 in P
          proof
            assume p2 in P;
            then ex q4 st q4=p2 & q4<>p & q4 in R & not ex P1 being Subset of
            TOP-REAL 2 st P1 is_S-P_arc_joining p,q4 & P1 c=R by A2;
            hence contradiction by A6,A7,A8,A14,Th23;
          end;
          hence contradiction by A4;
        end;
      end;
      hence contradiction;
    end;
  end;
  then PP is open by Lm1,TOPMETR:15;
  hence thesis by PRE_TOPC:30;
end;
