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 Th6:
  p`1 = q`1 & p`2 <> q`2 & p in Ball(u,r) & q in Ball(u,r) & f =<*
p,|[p`1,(p`2+q`2)/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 and
A2: p`2 <> q`2 and
A3: p in Ball(u,r) & q in Ball(u,r) and
A4: f =<* p,|[p`1,(p`2+q`2)/2]|,q *>;
  thus
A5: f is being_S-Seq & f/.1 =p & f/.len f = q by A1,A2,A4,TOPREAL3:36;
  p=|[p`1,p`2]| & q=|[q`1,q`2]| by EUCLID:53;
  then |[p`1,(p`2+q`2)/2]| in Ball(u,r) by A1,A3,TOPREAL3:23;
  then
A6: LSeg(p,|[p`1,(p`2+q`2)/2]|) c= Ball(u,r) & LSeg(|[p`1,(p`2+q`2)/2]|,q)
  c= Ball(u,r) by A3,TOPREAL3:21;
  thus L~f is_S-P_arc_joining p,q by A5;
  L~f=LSeg(p,|[p`1,(p`2+q`2)/2]|) \/ LSeg(|[p`1,(p`2+q`2)/2]|,q) by A4,
TOPREAL3:16;
  hence thesis by A6,XBOOLE_1:8;
end;
