reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th48:
  for f being S-Sequence_in_R2 st p in LSeg(f,i) & not p in rng f
  holds Ins(f,i,p) is being_S-Seq
proof
  let f be S-Sequence_in_R2 such that
A1: p in LSeg(f,i) and
A2: not p in rng f;
  set g = Ins(f,i,p);
  thus g is one-to-one by A2,FINSEQ_5:76;
  len g = len f + 1 by FINSEQ_5:69;
  then
A3: len g >= len f by NAT_1:11;
  len f >= 2 by TOPREAL1:def 8;
  hence len g >= 2 by A3,XXREAL_0:2;
  thus g is unfolded by A1,Th32;
  thus g is s.n.c. by A1,A2,Th37;
  thus thesis by A1,Th41;
end;
