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 Th39:
  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 f^mid(g,2,len g)
  is_S-Seq_joining f/.1,g/.len g
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: f^mid(g,2,len g) is being_S-Seq by A1,A2,A3,A4,Th38;
A6: len g >= 2 by A3,TOPREAL1:def 8;
  then
A7: 1+1-1<=len g-1 by XREAL_1:9;
  len f >= 2 by A2,TOPREAL1:def 8;
  then
A8: 1<=len f by XXREAL_0:2;
  then
A9: (f^mid(g,2,len g)).1=f.1 by FINSEQ_1:64
    .=f/.1 by A8,FINSEQ_4:15;
A10: len (f^mid(g,2,len g))=len f + len mid(g,2,len g) by FINSEQ_1:22;
A11: 1<=len g by A6,XXREAL_0:2;
  then
A12: len mid(g,2,len g)=len g -'2+1 by A6,FINSEQ_6:118;
  then
A13: len mid(g,2,len g) =len g -2+1 by A6,XREAL_1:233
    .=len g -1;
  then
A14: len (mid(g,2,len g))+2-1=len g;
  len g-1>=1+1-1 by A6,XREAL_1:9;
  then len f+1<=len (f^mid(g,2,len g)) by A10,A13,XREAL_1:6;
  then len f<len (f^mid(g,2,len g)) by NAT_1:13;
  then (f^mid(g,2,len g)).(len (f^mid(g,2,len g))) =(mid(g,2,len g)).(len (f^
  mid(g,2,len g))-len f) by FINSEQ_6:108
    .=g.len g by A6,A10,A12,A7,A14,FINSEQ_6:122
    .=g/.len g by A11,FINSEQ_4:15;
  hence thesis by A5,A9;
end;
