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 Th23:
  f1 is non empty & f2 is non empty implies L~(f1^f2) = L~f1 \/
  LSeg(f1/.len f1,f2/.1) \/ L~f2
proof
  set p = f1/.len f1, q = f2/.1;
  defpred P[FinSequence of TOP-REAL 2] means L~(f1^<*q*>^$1) = L~f1 \/ LSeg(p,
  q) \/ L~(<*q*>^$1);
  assume
A1: f1 is non empty;
A2: for f for o being Point of TOP-REAL 2 st P[f] holds P[f^<*o*>]
  proof
    let f;
    let o be Point of TOP-REAL 2 such that
A3: P[f];
    per cases;
    suppose
      f is empty;
      then
A4:   (f^<*o*>) = <*o*> by FINSEQ_1:34;
      len(f1^<*q*>) = len f1 + 1 by FINSEQ_2:16;
      then (f1^<*q*>)/.len(f1^<*q*>) = q by FINSEQ_4:67;
      hence L~(f1^<*q*>^(f^<*o*>)) = L~(f1^<*q*>) \/ LSeg(q,o) by A4,Th19
        .= L~f1 \/ LSeg(p,q) \/ LSeg(q,o) by A1,Th19
        .= L~f1 \/ LSeg(p,q) \/ L~<*q,o*> by Th21
        .= L~f1 \/ LSeg(p,q) \/ L~(<*q*>^(f^<*o*>)) by A4,FINSEQ_1:def 9;
    end;
    suppose
A5:   f is non empty;
      set g = f1^<*q*>^f, h = <*q*>^f;
      len f <> 0 by A5;
      then consider
      f9 being FinSequence of TOP-REAL 2, d being Point of TOP-REAL 2
      such that
A6:   f = f9^<*d*> by FINSEQ_2:19;
A7:   h = <*q*>^f9^<*d*> by A6,FINSEQ_1:32;
      then len h = len(<*q*>^f9)+1 by FINSEQ_2:16;
      then
A8:   h/.len h = d by A7,FINSEQ_4:67;
A9:   g = f1^<*q*>^f9^<*d*> by A6,FINSEQ_1:32;
      then len g = len(f1^<*q*>^f9)+1 by FINSEQ_2:16;
      then g/.len g = d by A9,FINSEQ_4:67;
      then
A10:  L~h \/ LSeg(g/.len g,o) = L~(h^<*o*>) by A8,Th19
        .= L~(<*q*>^(f^<*o*>)) by FINSEQ_1:32;
      thus L~(f1^<*q*>^(f^<*o*>)) = L~(g^<*o*>) by FINSEQ_1:32
        .= L~g \/ LSeg(g/.len g,o) by Th19
        .= L~f1 \/ LSeg(p,q) \/ L~(<*q*>^(f^<*o*>)) by A3,A10,XBOOLE_1:4;
    end;
  end;
  assume f2 is non empty;
  then
A11: f2 = <*q*>^(f2/^1) by FINSEQ_5:29;
A12: P[<*>(the carrier of TOP-REAL 2)]
  proof
    set O = <*>(the carrier of TOP-REAL 2);
    thus L~(f1^<*q*>^O) = L~(f1^<*q*>) by FINSEQ_1:34
      .= L~f1 \/ LSeg(p,q) \/ {} by A1,Th19
      .= L~f1 \/ LSeg(p,q) \/ L~<*q*> by Th12
      .= L~f1 \/ LSeg(p,q) \/ L~(<*q*>^O) by FINSEQ_1:34;
  end;
  for f holds P[f] from FINSEQ_2:sch 2(A12,A2);
  then L~(f1^<*q*>^(f2/^1)) = L~f1 \/ LSeg(p,q) \/ L~(<*q*>^(f2/^1));
  hence thesis by A11,FINSEQ_1:32;
end;
