reserve A for non empty closed_interval Subset of REAL;

theorem
for a,b,c be Real st a < b holds
['a,b'] c= [#]REAL & lower_bound ['a,b'] = a & upper_bound ['a,b'] = b
proof
let a,b,c be Real;
 assume A1: a < b;
 R1: [#]REAL = REAL by SUBSET_1:def 3;
 ['a,b'] = [.(lower_bound ['a,b']),(upper_bound ['a,b']).]
 & ['a,b'] =[.a,b.] by A1,INTEGRA5:def 3,INTEGRA1:4;
 hence thesis by INTEGRA1:5,R1;
end;
