reserve k,n,i,j for Nat;

theorem Th6:
  for D being non empty set, f being FinSequence of D st 2<= len f
  holds f = (f|(len f-'2))^mid(f,len f-'1,len f)
proof
  let D be non empty set, f be FinSequence of D;
  assume
A1: 2<= len f;
  then
A2: len f-'2=len f-2 by XREAL_1:233;
  then
A3: len f-'2+1=len f-1-1+1 .=len f-'1 by A1,XREAL_1:233,XXREAL_0:2;
  now
    per cases;
    case
      len f-'2>0;
      then
A4:   0+1<=len f-'2 by NAT_1:13;
      len f<len f+1 by NAT_1:13;
      then len f<len f+1+1 by NAT_1:13;
      then len f -2<len f+2-2 by XREAL_1:14;
      then f=mid(f,1,len f-'2)^mid(f,len f-'2+1,len f) by A2,A4,Th5;
      hence thesis by A3,A4,FINSEQ_6:116;
    end;
    case
A5:   len f-'2=0;
      then
A6:   mid(f,len f-'2+1,len f)=f by A1,FINSEQ_6:120,XXREAL_0:2;
A7:   f|0 is empty;
      len f-'2+1=(len f)-2+1 by A1,XREAL_1:233
        .=len f-1
        .=len f-'1 by A1,XREAL_1:233,XXREAL_0:2;
      hence thesis by A5,A6,A7,FINSEQ_1:34;
    end;
  end;
  hence thesis;
end;
