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
  R is being_Region & p in R & q in R & p<>q implies ex P st P
  is_S-P_arc_joining p,q & P c=R
proof
  set RR={q2: q2=p or ex P1 be Subset of TOP-REAL 2 st P1 is_S-P_arc_joining p
  ,q2 & P1 c=R};
  RR c= the carrier of TOP-REAL 2
  proof
    let x be object;
    assume x in RR;
    then ex q2 st q2=x & (q2=p or ex P1 be Subset of TOP-REAL 2 st P1
    is_S-P_arc_joining p,q2 & P1 c=R);
    hence thesis;
  end;
  then reconsider RR as Subset of TOP-REAL 2;
  assume that
A1: R is being_Region & p in R and
A2: q in R and
A3: p<>q;
  R c= RR by A1,Th27;
  then q in RR by A2;
  then ex q1 st q1=q & (q1=p or ex P1 be Subset of TOP-REAL 2 st P1
  is_S-P_arc_joining p,q1 & P1 c=R);
  hence thesis by A3;
end;
