reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;

theorem Th2:
  A = [. lower_bound A, upper_bound A .]
proof
  consider a,b being Real such that
A1: a <= b and
A2: A=[.a,b.] by MEASURE5:14;
A3: for y be Real st 0<y ex x be Real st x in A & b-y<x
  proof
    let y be Real;
    assume
A4: 0<y;
    take b;
    b < b+y by A4,XREAL_1:29;
    then b-y < b+y-y by XREAL_1:9;
    hence thesis by A1,A2,XXREAL_1:1;
  end;
A5: for x be Real st x in A holds x <= b
  proof
    let x be Real;
    assume
A6: x in A;
    A={y:a <= y & y <= b} by A2,RCOMP_1:def 1;
    then ex y st x=y & a <= y & y <= b by A6;
    hence thesis;
  end;
A7: for x be Real st x in A holds a <= x
  proof
    let x be Real;
    assume
A8: x in A;
    A={y:a <= y & y <= b} by A2,RCOMP_1:def 1;
    then ex y st x=y & a <= y & y <= b by A8;
    hence thesis;
  end;
  for y be Real st 0<y ex x be Real st x in A & x<a + y
  proof
    let y be Real;
    assume
A9: 0<y;
    take a;
    thus thesis by A1,A2,A9,XREAL_1:29,XXREAL_1:1;
  end;
  then a=lower_bound A by A7,SEQ_4:def 2;
  hence thesis by A2,A5,A3,SEQ_4:def 1;
end;
