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 Th24:
  q in rng f implies L~f = L~(f-:q) \/ L~(f:-q)
proof
  set n = q..f;
  assume
A1: q in rng f;
  then
A2: n <= len f by FINSEQ_4:21;
  per cases by A2,XXREAL_0:1;
  suppose
A3: n < len f;
    then len(f/^n) = len f - n by RFINSEQ:def 1;
    then len(f/^n) <> 0 by A3;
    then
A4: f/^n is non empty;
A5: len(f|n) = n by A3,FINSEQ_1:59;
    f|n = f-:q by FINSEQ_5:def 1;
    then
A6: (f|n)/.len(f|n) = q by A1,A5,FINSEQ_5:45;
A7: (f|n)^(f/^n) = f by RFINSEQ:8;
    f|n is non empty by A1,A5,FINSEQ_4:21;
    hence L~f = L~(f|n) \/ LSeg((f|n)/.len(f|n),(f/^n)/.1) \/ L~(f/^n) by A4,A7
,Th23
      .= L~(f|n) \/ (LSeg((f|n)/.len(f|n),(f/^n)/.1) \/ L~(f/^n)) by XBOOLE_1:4
      .= L~(f|n) \/ L~(<*(f|n)/.len(f|n)*>^(f/^n)) by A4,Th20
      .= L~(f|n) \/ L~(f:-q) by A6,FINSEQ_5:def 2
      .= L~(f-:q) \/ L~(f:-q) by FINSEQ_5:def 1;
  end;
  suppose
A8: n = len f;
    then len(f/^n) = len f - n by RFINSEQ:def 1
      .= 0 by A8;
    then
A9: f/^n is empty;
    f:-q = <*q*>^(f/^n) by FINSEQ_5:def 2
      .= <*q*> by A9,FINSEQ_1:34;
    then
A10: L~(f:-q) is empty by Th12;
    L~f = L~(f|n) by A8,FINSEQ_1:58
      .= L~(f-:q) by FINSEQ_5:def 1;
    hence thesis by A10;
  end;
end;
