reserve x,y for object,X for set,
  f for Function,
  R,S for Relation;
reserve e1,e2 for ExtReal;
reserve s,s1,s2,s3 for sequence of X;
reserve XX for non empty set,
        ss,ss1,ss2,ss3 for sequence of XX;

theorem Th21:
  for ss1 being subsequence of ss holds rng ss1 c= rng ss
proof
  let ss1 be subsequence of ss;
  let x be object;
  consider N being increasing sequence of NAT such that
A1: ss1 = ss * N by Def13;
    assume x in rng ss1;
    then consider n being object such that
A2: n in NAT and
A3: x = ss1.n by FUNCT_2:11;
A4: N.n in NAT by A2,FUNCT_2:5;
    x = ss.(N.n) by A1,A2,A3,FUNCT_2:15;
    hence thesis by A4,FUNCT_2:4;
end;
