
theorem Th39:
for A,B be Subset of REAL, F be Interval_Covering of A,
 G be Interval_Covering of B st F = G holds F vol = G vol
proof
    let A,B be Subset of REAL, F be Interval_Covering of A,
    G be Interval_Covering of B;
    assume
A1:  F = G;
    for n be Element of NAT holds (F vol).n = (G vol).n
    proof
     let n be Element of NAT;
     (F vol).n = diameter(F.n) by MEASURE7:def 4;
     hence (F vol).n = (G vol).n by A1,MEASURE7:def 4;
    end;
    hence F vol = G vol by FUNCT_2:def 8;
end;
