
theorem Th31:
for F be sequence of bool REAL, G be Open_Interval_Covering of F,
 H be sequence of [:NAT,NAT:] st rng H = [:NAT,NAT:] holds
  On(G,H) is Open_Interval_Covering of union rng F
proof
    let F be sequence of bool REAL, G be Open_Interval_Covering of F,
    H be sequence of [:NAT,NAT:];
    assume A1: rng H = [:NAT,NAT:];
    for n be Element of NAT holds On(G,H).n is open_interval
    proof
     let n be Element of NAT;
     On(G,H).n = (G.(pr1(H).n)).(pr2(H).n) by A1,MEASURE7:def 11;
     hence thesis;
    end;
    hence On(G,H) is Open_Interval_Covering of union rng F by Def5;
end;
