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 Th16:
  f is being_S-Seq & i>2 & i in dom f & h = f|i implies h is
being_S-Seq & h/.1=f/.1 & h/.len h=f/.i & L~h is_S-P_arc_joining f/.1,f/.i & L~
  h c= L~f & L~h = L~(f|i) \/ LSeg(f/.i,f/.i)
proof
  assume that
A1: f is being_S-Seq & i>2 and
A2: i in dom f and
A3: h = f|i;
A4: Seg len f = dom f by FINSEQ_1:def 3;
  then i<=len f by A2,FINSEQ_1:1;
  then
A5: Seg i c= Seg len f by FINSEQ_1:5;
  h = f|Seg i by A3,FINSEQ_1:def 16;
  then dom h=Seg(len f) /\ Seg i by A4,RELAT_1:61;
  then
A6: dom h = Seg i by A5,XBOOLE_1:28;
  1<=i by A2,A4,FINSEQ_1:1;
  then
A7: 1 in dom h & i in dom h by A6,FINSEQ_1:1;
 i in NAT by ORDINAL1:def 12;
  then len h = i by A6,FINSEQ_1:def 3;
  hence h is being_S-Seq & h/.1=f/.1 & h/.len h = f/.i by A1,A2,A3,A7,
FINSEQ_4:70,TOPREAL3:33;
  hence L~h is_S-P_arc_joining f/.1,f/.i & L~h c= L~f by A3,TOPREAL3:20;
  then f/.i in L~(f|i) by A3,Th3;
  then LSeg(f/.i,f/.i) = {f/.i} & {f/.i} c= L~(f|i) by RLTOPSP1:70,ZFMISC_1:31;
  hence thesis by A3,XBOOLE_1:12;
end;
