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 Th19:
  f is non empty implies L~(f^<*p*>) = L~f \/ LSeg(f/.len f,p)
proof
  set fp = f^<*p*>;
A1: len f + 1 <= len fp by FINSEQ_2:16;
  1 <= len f + 1 by NAT_1:11;
  then
A2: len f + 1 in dom fp by A1,FINSEQ_3:25;
A3: fp/.(len f + 1) = p by FINSEQ_4:67;
  len fp = len f + 1 by FINSEQ_2:16;
  then
A4: fp|(len f + 1) = fp by FINSEQ_1:58;
A5: dom f c= dom(fp) by FINSEQ_1:26;
A6: fp|len f = f by FINSEQ_5:23;
  assume f is non empty;
  then
A7: len f in dom f by FINSEQ_5:6;
  then fp/.len f = f/.len f by FINSEQ_4:68;
  hence thesis by A2,A7,A3,A4,A5,A6,TOPREAL3:38;
end;
