reserve a,b,r,x,y for Real,
  i,j,k,n for Nat,
  x1 for set;

theorem
  for A be non empty closed_interval Subset of REAL, x being Real holds
  x in A iff lower_bound A <= x & x <= upper_bound A
proof
  let A be non empty closed_interval Subset of REAL;
  let x be Real;
  hereby
    assume x in A;
    then x in [.lower_bound A,upper_bound A.] by INTEGRA1:4;
    then x in {a: lower_bound A <= a & a <= upper_bound A} by RCOMP_1:def 1;
    then ex a st a=x & lower_bound A <= a & a <= upper_bound A;
    hence lower_bound A <= x & x <= upper_bound A;
  end;
  assume
A1: lower_bound A <= x & x <= upper_bound A;
  x in {a: lower_bound A <= a & a <= upper_bound A} by A1;
  then x in [.lower_bound A,upper_bound A.] by RCOMP_1:def 1;
  hence thesis by INTEGRA1:4;
end;
