reserve X for non empty set,
  F for with_the_same_dom Functional_Sequence of X, ExtREAL,
  seq,seq1,seq2 for ExtREAL_sequence,
  x for Element of X,
  a,r for R_eal,
  n,m,k for Nat;

theorem Th1:
  (for n be Nat holds seq1.n <= seq2.n) implies inf rng seq1 <= inf rng seq2
proof
  assume
A1: for n be Nat holds seq1.n <= seq2.n;
  now
    let x be ExtReal;
A2: now
      let n be Element of NAT;
      dom seq1 = NAT by FUNCT_2:def 1;
      then
A3:   seq1.n in rng seq1 by FUNCT_1:def 3;
A4:   seq1.n <= seq2.n by A1;
      inf rng seq1 is LowerBound of rng seq1 by XXREAL_2:def 4;
      then inf rng seq1 <= seq1.n by A3,XXREAL_2:def 2;
      hence inf rng seq1 <= seq2.n by A4,XXREAL_0:2;
    end;
    assume x in rng seq2;
    then ex z be object st z in dom seq2 & x=seq2.z by FUNCT_1:def 3;
    hence inf rng seq1 <= x by A2;
  end;
  then inf rng seq1 is LowerBound of rng seq2 by XXREAL_2:def 2;
  hence thesis by XXREAL_2:def 4;
end;
