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 Th30:
  f is unfolded & k+1 = len f & LSeg(f,k) /\ LSeg(f/.len f,p) = {f
  /.len f} implies f^<*p*> is unfolded
proof
  set g = <*p*>;
  assume that
A1: f is unfolded and
A2: k+1 = len f and
A3: LSeg(f,k) /\ LSeg(f/.len f,p) = {f/.len f};
A4: len g = 1 by FINSEQ_1:39;
A5: g/.1 = p by FINSEQ_4:16;
A6: len(f^g) = len f + len g by FINSEQ_1:22;
  let i be Nat such that
A7: 1 <= i and
A8: i + 2 <= len(f^g);
A9: i+(1+1) = i+1+1;
  per cases;
  suppose
A10: i+2 <= len f;
    then
A11: i+1 in dom f by A7,SEQ_4:135;
    i+1 <= i+1+1 by NAT_1:11;
    hence LSeg(f^g,i) /\ LSeg(f^g,i+1) = LSeg(f,i) /\ LSeg(f^g,i+1) by A10,Th6,
XXREAL_0:2
      .= LSeg(f,i) /\ LSeg(f,i+1) by A9,A10,Th6
      .= {f/.(i+1)} by A1,A7,A10
      .= {(f^g)/.(i+1)} by A11,FINSEQ_4:68;
  end;
  suppose
    i + 2 > len f;
    then
A12: len f <= i+1 by A9,NAT_1:13;
A13: f is non empty by A4,A8,A6,A9,XREAL_1:6;
    then
A14: len f in dom f by FINSEQ_5:6;
    i+1 <= len f by A4,A8,A6,A9,XREAL_1:6;
    then
A15: i+1 = len f by A12,XXREAL_0:1;
    then
A16: LSeg(f^g,i+1) = LSeg(f/.len f,g/.1) by A13,Th8;
    LSeg(f^g,i) = LSeg(f,k) by A2,A15,Th6;
    hence thesis by A3,A5,A15,A16,A14,FINSEQ_4:68;
  end;
end;
