reserve k,n,i,j for Nat;

theorem Th7:
  for D being non empty set, f being FinSequence of D st 1<= len f
  holds f = (f|(len f-'1))^mid(f,len f,len f)
proof
  let D be non empty set, f be FinSequence of D;
  assume
A1: 1<= len f;
  then
A2: len f-'1=len f-1 by XREAL_1:233;
  now
    per cases;
    case
      len f-'1>0;
      then
A3:   0+1<=len f-'1 by NAT_1:13;
      len f<len f+1 by NAT_1:13;
      then len f -1<len f+1-1 by XREAL_1:14;
      then f=mid(f,1,len f-'1)^mid(f,len f-'1+1,len f) by A2,A3,Th5;
      hence thesis by A2,A3,FINSEQ_6:116;
    end;
    case
A4:   len f-'1=0;
A5:   f|0 is empty;
      mid(f,len f-'1+1,len f)=f by A1,A4,FINSEQ_6:120;
      hence thesis by A2,A4,A5,FINSEQ_1:34;
    end;
  end;
  hence thesis;
end;
