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 Th42:
  q in rng f & 1<>q..f & q..f<>len f & f is unfolded s.n.c.
  implies L~(f-:q) /\ L~(f:-q) = {q}
proof
  assume that
A1: q in rng f and
A2: 1 <> q..f and
A3: q..f <> len f and
A4: f is unfolded and
A5: f is s.n.c.;
A6: (f:-q)/.1 = q by FINSEQ_5:53;
  q..f <= len f by A1,FINSEQ_4:21;
  then q..f < len f by A3,XXREAL_0:1;
  then q..f+1 <= len f by NAT_1:13;
  then
A7: 1 <= len f - q..f by XREAL_1:19;
  len(f:-q) = len f - q..f + 1 by A1,FINSEQ_5:50;
  then 1+1<=len(f:-q) by A7,XREAL_1:6;
  then
A8: rng(f:-q) c= L~(f:-q) by Th18;
  1 in dom(f:-q) by FINSEQ_5:6;
  then
A9: q in rng(f:-q) by A6,PARTFUN2:2;
  f-:q is non empty by A1,FINSEQ_5:47;
  then
A10: len(f-:q) in dom(f-:q) by FINSEQ_5:6;
A11: len(f-:q) = q..f by A1,FINSEQ_5:42;
  thus L~(f-:q) /\ L~(f:-q) c= {q}
  proof
    let x be object;
    assume
A12: x in L~(f-:q) /\ L~(f:-q);
    then reconsider p = x as Point of TOP-REAL 2;
    p in L~(f-:q) by A12,XBOOLE_0:def 4;
    then consider i such that
A13: 1 <= i and
A14: i+1 <= len(f-:q) and
A15: p in LSeg((f-:q),i) by Th13;
A16: LSeg(f-:q,i) = LSeg(f,i) by A14,Th9;
    p in L~(f:-q) by A12,XBOOLE_0:def 4;
    then consider j such that
A17: 1 <= j and
    j+1 <= len(f:-q) and
A18: p in LSeg((f:-q),j) by Th13;
    consider j9 being Nat such that
A19: j=j9+1 by A17,NAT_1:6;
    reconsider j9 as Element of NAT by ORDINAL1:def 12;
A20: LSeg(f:-q,j) = LSeg(f,j9+q..f) by A1,A19,Th10;
    per cases;
    suppose that
A21:  i+1 = len(f-:q) and
A22:  j9 = 0;
      q..f <= len f by A1,FINSEQ_4:21;
      then q..f < len f by A3,XXREAL_0:1;
      then i+1+1 <= len f by A11,A21,NAT_1:13;
      then i+(1+1) <= len f;
      then LSeg((f-:q),i) /\ LSeg(f:-q,j) = {f/.(q..f)} by A4,A11,A13,A16,A20
,A21,A22
        .= {q} by A1,FINSEQ_5:38;
      hence thesis by A15,A18,XBOOLE_0:def 4;
    end;
    suppose that
A23:  i+1 = len(f-:q) and
A24:  j9 <> 0;
      1 <= j9 by A24,NAT_1:14;
      then i+1+1 <= j9+q..f by A11,A23,XREAL_1:7;
      then i+1 < j9+q..f by NAT_1:13;
      then LSeg((f-:q),i) misses LSeg(f:-q,j) by A5,A16,A20;
      then LSeg((f-:q),i) /\ LSeg(f:-q,j) = {};
      hence thesis by A15,A18,XBOOLE_0:def 4;
    end;
    suppose
A25:  i+1 <> len(f-:q);
A26:  q..f <= j9+q..f by NAT_1:11;
      i+1 < q..f by A11,A14,A25,XXREAL_0:1;
      then i+1 < j9+q..f by A26,XXREAL_0:2;
      then LSeg((f-:q),i) misses LSeg(f:-q,j) by A5,A16,A20;
      then LSeg((f-:q),i) /\ LSeg(f:-q,j) = {};
      hence thesis by A15,A18,XBOOLE_0:def 4;
    end;
  end;
  1 <= q..f by A1,FINSEQ_4:21;
  then 1 < q..f by A2,XXREAL_0:1;
  then 1+1 <= q..f by NAT_1:13;
  then
A27: rng(f-:q) c= L~(f-:q) by A11,Th18;
  (f-:q)/.(q..f) = q by A1,FINSEQ_5:45;
  then q in rng(f-:q) by A11,A10,PARTFUN2:2;
  then q in L~(f-:q) /\ L~(f:-q) by A27,A9,A8,XBOOLE_0:def 4;
  hence thesis by ZFMISC_1:31;
end;
