let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M holds
( a.e-eq-class f,M = a.e-eq-class g,M iff f a.e.= g,M )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M holds
( a.e-eq-class f,M = a.e-eq-class g,M iff f a.e.= g,M )
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M holds
( a.e-eq-class f,M = a.e-eq-class g,M iff f a.e.= g,M )
let f, g be PartFunc of X,REAL ; :: thesis: ( f in L1_Functions M & g in L1_Functions M implies ( a.e-eq-class f,M = a.e-eq-class g,M iff f a.e.= g,M ) )
assume that
A1: f in L1_Functions M and
A2: g in L1_Functions M ; :: thesis: ( a.e-eq-class f,M = a.e-eq-class g,M iff f a.e.= g,M )

hereby :: thesis: ( f a.e.= g,M implies a.e-eq-class f,M = a.e-eq-class g,M )

assume a.e-eq-class f,M = a.e-eq-class g,M ; :: thesis: f a.e.= g,M
then f in { r where r is PartFunc of X,REAL : ( r in L1_Functions M & g in L1_Functions M & g a.e.= r,M ) } by A1, Th38;
then ex r being PartFunc of X,REAL st
( f = r & r in L1_Functions M & g in L1_Functions M & g a.e.= r,M ) ;
hence f a.e.= g,M by Th29; :: thesis: verum

end;

assume A3: f a.e.= g,M ; :: thesis: a.e-eq-class f,M = a.e-eq-class g,M

now

let x be set ; :: thesis: ( x in a.e-eq-class f,M implies x in a.e-eq-class g,M )
assume x in a.e-eq-class f,M ; :: thesis: x in a.e-eq-class g,M
then consider r being PartFunc of X,REAL such that
A4: ( x = r & r in L1_Functions M ) and
f in L1_Functions M and
A5: f a.e.= r,M ;
g a.e.= f,M by A3, Th29;
then g a.e.= r,M by A5, Th30;
hence x in a.e-eq-class g,M by A2, A4; :: thesis: verum

end;

then A6: a.e-eq-class f,M c= a.e-eq-class g,M by TARSKI:def 3;

now

let x be set ; :: thesis: ( x in a.e-eq-class g,M implies x in a.e-eq-class f,M )
assume x in a.e-eq-class g,M ; :: thesis: x in a.e-eq-class f,M
then consider r being PartFunc of X,REAL such that
A7: ( x = r & r in L1_Functions M ) and
g in L1_Functions M and
A8: g a.e.= r,M ;
f a.e.= r,M by A3, A8, Th30;
hence x in a.e-eq-class f,M by A1, A7; :: thesis: verum

end;

then a.e-eq-class g,M c= a.e-eq-class f,M by TARSKI:def 3;
hence a.e-eq-class f,M = a.e-eq-class g,M by A6, XBOOLE_0:def 10; :: thesis: verum