
theorem Th30:
for A be Subset of REAL holds Svc2 A c= Svc A & inf Svc A <= inf Svc2 A
proof
    let A be Subset of REAL;
    now let x be R_eal;
     assume x in Svc2(A); then
     ex F be Open_Interval_Covering of A st x = vol F by Def7;
     hence x in Svc(A) by MEASURE7:def 8;
    end;
    hence Svc2 A c= Svc A;
    hence inf Svc A <= inf Svc2 A by XXREAL_2:60;
end;
