reserve a,b,c,d,e for Real;
reserve X,Y for set,
          Z for non empty set,
          r for Real,
          s for ExtReal,
          A for Subset of REAL,
          f for real-valued Function;
reserve I for non empty closed_interval Subset of REAL,
       TD for tagged_division of I,
        D for Division of I,
        T for Element of set_of_tagged_Division(D),
        f for PartFunc of I,REAL;
reserve f for Function of I,REAL;
reserve f,g for HK-integrable Function of I,REAL,
          r for Real;
reserve f for bounded integrable Function of I,REAL;
reserve jauge for positive-yielding Function of I,REAL;
reserve D for tagged_division of I;
reserve r1,r2,s for Real,
           D,D1 for Division of I,
             fc for Function of I,REAL;

theorem Th45:
  for p being FinSequence of REAL st
  (for i being Nat st i in dom p holds r <= p.i) &
  ex i0 being Nat st i0 in dom p & p.i0 = r holds
  r = inf rng p
  proof
    let p be FinSequence of REAL;
    assume that
A1: for i be Nat st i in dom p holds r <= p.i and
A2: ex i0 be Nat st i0 in dom p & p.i0 = r;
    set X = rng p;
    consider i0 be Nat such that
A3: i0 in dom p and
A4: p.i0 = r by A2;
    reconsider X as non empty bounded_below real-membered set
      by A3,A4,FUNCT_1:def 3;
A5: for a be Real st a in X holds r <= a
    proof
      let a be Real;
      assume a in X;
      then ex i be object st i in dom p & p.i = a by FUNCT_1:def 3;
      hence thesis by A1;
    end;
    for s be Real st 0 < s ex a be Real st a in X & a < r + s
    proof
      let s be Real;
      assume
A6:   0 < s;
      consider i0 be Nat such that
A7:   i0 in dom p and
A8:   p.i0 = r by A2;
      reconsider a = p.i0 as Real;
      take a;
      thus a in X by A7,FUNCT_1:def 3;
      r + 0 < r + s by A6,XREAL_1:8;
      hence a < r + s by A8;
    end;
    then r = lower_bound X by A5,SEQ_4:def 2;
    hence thesis;
  end;
