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 Th9:
  p`1 <> q`1 & p`2 <> q`2 & p in Ball(u,r) & q in Ball(u,r) & |[q
`1,p`2]| in Ball(u,r) & f =<* p,|[q`1,p`2]|,q *> implies f is being_S-Seq & f/.
  1 =p & f/.len f = q & L~f is_S-P_arc_joining p,q & L~f c= Ball(u,r)
proof
  assume that
A1: p`1 <> q`1 & p`2 <> q`2 and
A2: p in Ball(u,r) and
A3: q in Ball(u,r) and
A4: |[q`1,p`2]| in Ball(u,r) and
A5: f =<* p,|[q`1,p`2]|,q *>;
  thus
A6: f is being_S-Seq & f/.1 =p & f/.len f = q by A1,A5,TOPREAL3:35;
A7: LSeg(|[q`1,p`2]|,q) c= Ball(u,r) by A3,A4,TOPREAL3:21;
  thus L~f is_S-P_arc_joining p,q by A6;
  L~f=LSeg(p,|[q`1,p`2]|) \/ LSeg(|[q`1,p`2]|,q) & LSeg(p,|[q`1,p`2]|) c=
  Ball (u,r) by A2,A4,A5,TOPREAL3:16,21;
  hence thesis by A7,XBOOLE_1:8;
end;
