
theorem Th1:
  for D being set,p being FinSequence of D,i,j being Element of NAT
  holds rng Del(p,i,j) c= rng p
proof
  let D be set,p be FinSequence of D,i,j be Element of NAT;
A1: rng (p/^j) c= rng p
  proof
    per cases;
    suppose
A2:   D is empty;
then A3:  j>len p implies thesis;
     j<=len p implies thesis by A2;
     hence thesis by A3;
    end;
    suppose
      D is non empty;
      then reconsider E=D as non empty set;
      reconsider r=p as FinSequence of E;
      rng (r/^j) c= rng r by FINSEQ_5:33;
      hence thesis;
    end;
  end;
  rng (p|(i -' 1)) = rng (p|Seg(i -' 1)) by FINSEQ_1:def 16;
  then
A4: rng (p|(i -' 1)) c= rng p by RELAT_1:70;
  rng ((p|(i -' 1))^(p/^j)) = (rng (p|(i -' 1))) \/ rng (p/^j) by FINSEQ_1:31;
  hence thesis by A4,A1,XBOOLE_1:8;
end;
