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;

theorem
  P is_S-P_arc_joining p,q implies p<>q
proof
  assume that
A1: P is_S-P_arc_joining p,q and
A2: p=q;
  consider f such that
A3: f is being_S-Seq and
  P = L~f and
A4: p=f/.1 & q=f/.len f by A1;
  len f >= 2 by A3;
  then Seg len f = dom f & len f >= 1 by FINSEQ_1:def 3,XXREAL_0:2;
  then
A5: len f in dom f & 1 in dom f by FINSEQ_1:1;
  f is one-to-one & 1 <> len f by A3;
  hence contradiction by A2,A4,A5,PARTFUN2:10;
end;
