let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g, f1, g1 being PartFunc of X,COMPLEX st f in L1_CFunctions M & f1 in L1_CFunctions M & g in L1_CFunctions M & g1 in L1_CFunctions M & a.e-Ceq-class (f,M) = a.e-Ceq-class (f1,M) & a.e-Ceq-class (g,M) = a.e-Ceq-class (g1,M) holds
a.e-Ceq-class ((f + g),M) = a.e-Ceq-class ((f1 + g1),M)
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g, f1, g1 being PartFunc of X,COMPLEX st f in L1_CFunctions M & f1 in L1_CFunctions M & g in L1_CFunctions M & g1 in L1_CFunctions M & a.e-Ceq-class (f,M) = a.e-Ceq-class (f1,M) & a.e-Ceq-class (g,M) = a.e-Ceq-class (g1,M) holds
a.e-Ceq-class ((f + g),M) = a.e-Ceq-class ((f1 + g1),M)
let M be sigma_Measure of S; :: thesis: for f, g, f1, g1 being PartFunc of X,COMPLEX st f in L1_CFunctions M & f1 in L1_CFunctions M & g in L1_CFunctions M & g1 in L1_CFunctions M & a.e-Ceq-class (f,M) = a.e-Ceq-class (f1,M) & a.e-Ceq-class (g,M) = a.e-Ceq-class (g1,M) holds
a.e-Ceq-class ((f + g),M) = a.e-Ceq-class ((f1 + g1),M)
let f, g, f1, g1 be PartFunc of X,COMPLEX; :: thesis: ( f in L1_CFunctions M & f1 in L1_CFunctions M & g in L1_CFunctions M & g1 in L1_CFunctions M & a.e-Ceq-class (f,M) = a.e-Ceq-class (f1,M) & a.e-Ceq-class (g,M) = a.e-Ceq-class (g1,M) implies a.e-Ceq-class ((f + g),M) = a.e-Ceq-class ((f1 + g1),M) )
assume that
A1: ( f in L1_CFunctions M & f1 in L1_CFunctions M & g in L1_CFunctions M & g1 in L1_CFunctions M ) and
A2: ( a.e-Ceq-class (f,M) = a.e-Ceq-class (f1,M) & a.e-Ceq-class (g,M) = a.e-Ceq-class (g1,M) ) ; :: thesis: a.e-Ceq-class ((f + g),M) = a.e-Ceq-class ((f1 + g1),M)
( f a.e.cpfunc= f1,M & g a.e.cpfunc= g1,M ) by A1, A2, Th32;
then A3: f + g a.e.cpfunc= f1 + g1,M by Th25;
( f + g in L1_CFunctions M & f1 + g1 in L1_CFunctions M ) by A1, Th17;
hence a.e-Ceq-class ((f + g),M) = a.e-Ceq-class ((f1 + g1),M) by A3, Th32; :: thesis: verum