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 Th41:
  f is special & p in LSeg(f,n) implies Ins(f,n,p) is special
proof
  assume that
A1: f is special and
A2: p in LSeg(f,n);
A3: n+1 <= len f by A2,TOPREAL1:def 3;
  then
A4: 1 <= len f - n by XREAL_1:19;
A5: 1 <= n by A2,TOPREAL1:def 3;
  then
A6: LSeg(f,n) = LSeg(f/.n,f/.(n+1)) by A3,TOPREAL1:def 3;
  set f1 = f|n, g1 = f1^<*p*>, f2 = f/^n;
  set p1 = f1/.len f1, p2 = f2/.1;
A7: p1 = |[p1`1, p1`2]| by EUCLID:53;
A8: n <= n+1 by NAT_1:11;
  then n <= len f by A3,XXREAL_0:2;
  then 1 <= len f2 by A4,RFINSEQ:def 1;
  then 1 in dom f2 by FINSEQ_3:25;
  then
A9: f/.(n+1) = f2/.1 by FINSEQ_5:27;
A10: len f1 = n by A3,A8,FINSEQ_1:59,XXREAL_0:2;
  then n in dom f1 by A5,FINSEQ_3:25;
  then
A11: f/.n = f1/.len f1 by A10,FINSEQ_4:70;
  then
A12: p1`1 = p2`1 or p1`2 = p2`2 by A1,A5,A3,A9;
  set q1 = g1/.len g1;
A13: p2 = |[p2`1, p2`2]| by EUCLID:53;
  <*p*>/.1 = p by FINSEQ_4:16;
  then p1`1 = (<*p*>/.1)`1 or p1`2 = (<*p*>/.1)`2 by A2,A6,A11,A9,A12,A7,A13,
TOPREAL3:11,12;
  then
A14: g1 is special by A1,Lm13;
  g1/.len g1 = g1/.(len f1 + 1) by FINSEQ_2:16
    .= p by FINSEQ_4:67;
  then q1`1 = p2`1 or q1`2 = p2`2 by A2,A6,A11,A9,A12,A7,A13,TOPREAL3:11,12;
  then g1^f2 is special by A1,A14,Lm13;
  hence thesis by FINSEQ_5:def 4;
end;
