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 Th46:
  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_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: len f-'1<=len f by NAT_D:50;
A6: len f >= 2 by A2,TOPREAL1:def 8;
  then 1+1-1<=len f-1 by XREAL_1:9;
  then
A7: 1<=len f-'1 by NAT_D:39;
A8: 1<=len f by A6,XXREAL_0:2;
  then len mid(f,1,len f-'1)=len f-'1-'1+1 by A5,A7,FINSEQ_6:118
    .=len f-'1 -1+1 by A7,XREAL_1:233
    .=len f -'1;
  then
A9: (mid(f,1,len f-'1)^g).1=mid(f,1,len f-'1).1 by A7,FINSEQ_1:64
    .=f.1 by A5,A7,FINSEQ_6:123
    .=f/.1 by A8,FINSEQ_4:15;
A10: len (mid(f,1,len f-'1)^g)=len (mid(f,1,len f-'1)) + len g by FINSEQ_1:22;
A11: len g >= 2 by A3,TOPREAL1:def 8;
  then
A12: 1<=len g by XXREAL_0:2;
  0+len (mid(f,1,len f-'1))<len g+len (mid(f,1,len f-'1)) by A11,XREAL_1:6;
  then
A13: (mid(f,1,len f-'1)^g).(len (mid(f,1,len f-'1)^g)) =g.(len (mid(f,1,len
  f-'1)^g)-len (mid(f,1,len f-'1))) by A10,FINSEQ_6:108
    .=g/.len g by A12,A10,FINSEQ_4:15;
  mid(f,1,len f-'1)^g is being_S-Seq by A1,A2,A3,A4,Th45;
  hence thesis by A9,A13;
end;
