
theorem
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f,g be PartFunc of X,REAL st f a.e.= g,M holds f a.e.= g,COM M
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f,g be PartFunc of X,REAL;
    assume f a.e.= g,M; then
    consider E be Element of S such that
A1:  M.E = 0 & f|E` = g|E` by LPSPACE1:def 10;

    reconsider E0 = {} as Element of S by MEASURE1:7;
    M.E0 = 0 by VALUED_0:def 19; then
A2: E0 is thin of M by MEASURE3:def 2;

A3: E = E \/ E0;
    reconsider E1 = E as Element of COM(S,M) by Th27;

    (COM M).E1 = 0 by A1,A2,A3,MEASURE3:def 5;
    hence f a.e.= g,COM M by A1,LPSPACE1:def 10;
end;
