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 Th26:
  p in R & P={q: q=p or ex P1 being Subset of TOP-REAL 2 st P1
  is_S-P_arc_joining p,q & P1 c=R} implies P c= R
proof
  assume that
A1: p in R and
A2: P = {q: q=p or ex P1 being Subset of TOP-REAL 2 st P1
  is_S-P_arc_joining p,q & P1 c=R};
  let x be object;
  assume x in P;
  then consider q such that
A3: q=x and
A4: q=p or ex P1 being Subset of TOP-REAL 2 st P1 is_S-P_arc_joining p,q
  & P1 c=R by A2;
  now
    per cases by A4;
    suppose
      q=p;
      hence thesis by A1,A3;
    end;
    suppose
      ex P1 being Subset of TOP-REAL 2 st P1 is_S-P_arc_joining p,q & P1 c=R;
      then consider P1 being Subset of TOP-REAL 2 such that
A5:   P1 is_S-P_arc_joining p,q and
A6:   P1 c=R;
      q in P1 by A5,Th3;
      hence thesis by A3,A6;
    end;
  end;
  hence thesis;
end;
