
theorem
  for D being non empty set,f being FinSequence of D
  holds ovlcon(f,f)=f & ovlpart(f,f)=f & ovlldiff(f,f)={} & ovlrdiff(f,f)={}
proof
  let D be non empty set,f be FinSequence of D;
A1: ovlpart(f,f)=smid(f,1,len ovlpart(f,f)) by Def2;
  len f-'len f+1=len f-len f+1 by XREAL_1:233
    .=0+1;
  then smid(f,1,len f)=smid(f,len f-'len f+1,len f);
  then
A2: len f<=len ovlpart(f,f) by Def2;
  then
A3: ovlcon(f,f)=f^(<*>D) by Th2
    .=f by FINSEQ_1:34;
A4: ovlldiff(f,f)= f|(len f-'len f) by A1,A2,Th6
    .= f|(0) by XREAL_1:232
    .= {};
  ovlrdiff(f,f)= {} by A2,Th2;
  hence thesis by A1,A2,A3,A4,Th6;
end;
