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 Th11:
  p<>f/.1 & (f/.1)`2 = p`2 & f is being_S-Seq & p in LSeg(f,1) & h
= <* f/.1,|[((f/.1)`1+p`1)/2,(f/.1)`2]|,p *> implies h is being_S-Seq & h/.1=f
  /.1 & h/.len h=p & L~h is_S-P_arc_joining f/.1,p & L~h c= L~f & L~h = L~(f|1)
  \/ LSeg(f/.1,p)
proof
  assume that
A1: p<>f/.1 and
A2: (f/.1)`2 = p`2 and
A3: f is being_S-Seq and
A4: p in LSeg(f,1) and
A5: h = <* f/.1,|[((f/.1)`1+p`1)/2, (f/.1)`2]|,p *>;
  set p1 = f/.1, q = f/.(1+1);
A6: L~h=LSeg(p1,|[(p1`1 + p`1)/2,p1`2]|) \/ LSeg(|[(p1`1 + p`1)/2,p1`2]|,p)
  by A5,TOPREAL3:16;
A7: len f >= 2 by A3;
  then
A8: LSeg(f,1) = LSeg(p1,q) by TOPREAL1:def 3;
A9: p1`1 <> p`1 by A1,A2,TOPREAL3:6;
  hence
A10: h is being_S-Seq & h/.1 = p1 & h/.len h = p by A2,A5,TOPREAL3:37;
  p1 in LSeg(p1,q) by RLTOPSP1:68;
  then
A11: LSeg(p1,p) c= LSeg(p1,q) by A4,A8,TOPREAL1:6;
A12: Seg len f = dom f by FINSEQ_1:def 3;
  thus L~h is_S-P_arc_joining p1,p by A10;
A13: LSeg(f,1) c= L~f by TOPREAL3:19;
  LSeg(p1,|[(p1`1 + p`1)/2,p1`2]|) \/ LSeg(|[(p1`1 + p`1)/2,p1`2]|,p) =
  LSeg(p1,p) by A2,A9,TOPREAL1:5,TOPREAL3:13;
  hence L~h c= L~f by A8,A11,A6,A13;
  len f >= 1 by A7,XXREAL_0:2;
  then Seg 1 c= Seg len f by FINSEQ_1:5;
  then f|1 = f|Seg 1 & dom f /\ Seg 1 = Seg 1 by A12,FINSEQ_1:def 16
,XBOOLE_1:28;
  then dom(f|1) = Seg 1 by RELAT_1:61;
  then len (f|1) = 1 by FINSEQ_1:def 3;
  then L~(f|1)={} by TOPREAL1:22;
  hence thesis by A2,A9,A6,TOPREAL1:5,TOPREAL3:13;
end;
