reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;
reserve D for non empty set;

theorem
  for f being FinSequence,k1,k2 being Nat holds
  rng mid(f,k1,k2) c= rng f
proof
  let f be FinSequence,k1,k2 be Nat;
  per cases;
  suppose
    k1<=k2;
    then mid(f,k1,k2) = (f/^(k1-'1))|(k2-'k1+1) by Def3;
    then
A1: rng mid(f,k1,k2) c= rng (f/^(k1-'1)) by FINSEQ_5:19;
    rng (f/^(k1-'1)) c= rng f by FINSEQ_5:33;
    hence thesis by A1;
  end;
  suppose
    k1>k2;
    then mid(f,k1,k2) = Rev ((f/^(k2-'1))|(k1-'k2+1)) by Def3;
    then rng mid(f,k1,k2) = rng ((f/^(k2-'1))|(k1-'k2+1)) by FINSEQ_5:57;
    then
A2: rng mid(f,k1,k2) c= rng (f/^(k2-'1)) by FINSEQ_5:19;
    rng (f/^(k2-'1)) c= rng f by FINSEQ_5:33;
    hence thesis by A2;
  end;
end;
