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 Th45:
  for f,g being FinSequence of TOP-REAL 2 st f.len f=g.1 & f is
being_S-Seq & g is being_S-Seq & L~f /\ L~g={g.1} holds mid(f,1,len f-'1)^g is
  being_S-Seq
proof
  let f,g be FinSequence of TOP-REAL 2;
  assume that
A1: f.len f=g.1 and
A2: f is being_S-Seq and
A3: g is being_S-Seq and
A4: L~f /\ L~g={g.1};
A5: Rev f is being_S-Seq by A2;
  L~(Rev f)=L~f by SPPOL_2:22;
  then
A6: L~(Rev g)/\ L~(Rev f)={g.1} by A4,SPPOL_2:22;
A7: (Rev f).1=f.len f by FINSEQ_5:62;
A8: Rev g is being_S-Seq by A3;
  (Rev g).len (Rev g)=(Rev Rev g).1 by FINSEQ_5:62
    .=(Rev f).1 by A1,A7;
  then
A9: (Rev g)^(mid(Rev f,2,len (Rev f))) is being_S-Seq by A1,A5,A8,A6,A7,Th38;
A10: len f-'1<=len f by NAT_D:50;
A11: len (Rev f)=len f by FINSEQ_5:def 3;
A12: len f >= 2 by A2,TOPREAL1:def 8;
  then
A13: len f-1>=1+1-1 by XREAL_1:9;
A14: len f-'1+1=len f-1+1 by A12,XREAL_1:233,XXREAL_0:2
    .=len f;
A15: len f-'(len f-'1)+1=len f-(len f-'1)+1 by NAT_D:50,XREAL_1:233
    .=len f-(len f-1)+1 by A12,XREAL_1:233,XXREAL_0:2
    .=2;
  1<=len f by A12,XXREAL_0:2;
  then (Rev g)^(Rev mid(f,1,len f-'1)) is being_S-Seq by A13,A10,A15,A11,A14,A9
,FINSEQ_6:113;
  then Rev (mid(f,1,len f-'1)^g) is being_S-Seq by FINSEQ_5:64;
  then Rev Rev (mid(f,1,len f-'1)^g) is being_S-Seq;
  hence thesis;
end;
