
theorem Th32:
for A be Subset of REAL, G be sequence of bool REAL st
 A c= union rng G & (for n be Element of NAT holds G.n is open_interval)
 holds G is Open_Interval_Covering of A
proof
    let A be Subset of REAL, G be sequence of bool REAL;
    assume that
A1:  A c= union rng G and
A2:  for n be Element of NAT holds G.n is open_interval;
    now let n be Element of NAT;
     G.n is open_interval by A2;
     hence G.n is Interval;
    end; then
    G is Interval_Covering of A by A1,MEASURE7:def 2;
    hence G is Open_Interval_Covering of A by A2,Def5;
end;
