reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem Th43:
  for f,g being FinSequence of TOP-REAL 2, p being Point of
TOP-REAL 2 st f.len f=g.1 & p in L~f & f is being_S-Seq & g is being_S-Seq & L~
f /\ L~g={g.1} & p<>f.len f holds L_Cut(f,p)^mid(g,2,len g) is_S-Seq_joining p,
  g/.len g
proof
  let f,g be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2;
  assume that
A1: f.len f=g.1 and
A2: p in L~f and
A3: f is being_S-Seq and
A4: g is being_S-Seq and
A5: L~f /\ L~g={g.1} and
A6: p<>f.len f;
  L_Cut(f,p) is_S-Seq_joining p,f/.len f by A2,A3,A6,Th33;
  then
A7: L_Cut(f,p).(len L_Cut(f,p))=f/.len f;
A8: len g >= 2 by A4,TOPREAL1:def 8;
  then
A9: 1<=len g by XXREAL_0:2;
  g/.1 in LSeg(g/.1,g/.(1+1)) by RLTOPSP1:68;
  then g/.1 in LSeg(g,1) by A8,TOPREAL1:def 3;
  then g.1 in LSeg(g,1) by A9,FINSEQ_4:15;
  then
A10: g.1 in L~g by SPPOL_2:17;
  L~(L_Cut(f,p)) c= L~f by A2,Th42;
  then
A11: L~(L_Cut(f,p))/\ L~g c= L~f /\ L~g by XBOOLE_1:27;
  len f >= 2 by A3,TOPREAL1:def 8;
  then
A12: 1<=len f by XXREAL_0:2;
A13: L_Cut(f,p) is being_S-Seq by A2,A3,A6,Th34;
  then
A14: 1+1<=len L_Cut(f,p) by TOPREAL1:def 8;
  then
A15: 1+1-1<=len L_Cut(f,p)-1 by XREAL_1:9;
A16: 1<=len L_Cut(f,p) by A14,XXREAL_0:2;
  then L_Cut(f,p).1=(L_Cut(f,p))/.1 by FINSEQ_4:15;
  then
A17: (L_Cut(f,p))/.(1)=p by A2,Th23;
A18: len L_Cut(f,p)-'1+1=len L_Cut(f,p) by A14,XREAL_1:235,XXREAL_0:2;
  then
  (L_Cut(f,p))/.(len L_Cut(f,p)) in LSeg((L_Cut(f,p))/.(len L_Cut(f,p) -'
  1), (L_Cut(f,p))/.(len L_Cut(f,p)-'1+1)) by RLTOPSP1:68;
  then L_Cut(f,p).len L_Cut(f,p) in LSeg((L_Cut(f,p))/.(len L_Cut(f,p)-'1), (
  L_Cut(f,p))/.(len L_Cut(f,p)-'1+1)) by A16,FINSEQ_4:15;
  then
  L_Cut(f,p).len L_Cut(f,p) in LSeg(L_Cut(f,p),len L_Cut(f,p)-'1) by A15,A18,
TOPREAL1:def 3;
  then f/.len f in L~(L_Cut(f,p)) by A7,SPPOL_2:17;
  then f.len f in L~(L_Cut(f,p)) by A12,FINSEQ_4:15;
  then g.1 in L~(L_Cut(f,p))/\ L~g by A1,A10,XBOOLE_0:def 4;
  then {g.1}c= L~(L_Cut(f,p))/\ L~g by ZFMISC_1:31;
  then L~(L_Cut(f,p))/\ L~g={g.1} by A5,A11,XBOOLE_0:def 10;
  hence thesis by A1,A4,A12,A13,A7,A17,Th39,FINSEQ_4:15;
end;
