set C = CosetSet M;
let f1, f2 be Function of [:REAL,(CosetSet M):],(CosetSet M); :: thesis: ( ( for z being Real
for A being Element of CosetSet M
for f being PartFunc of X,REAL st f in A holds
f1 . (z,A) = a.e-eq-class ((z (#) f),M) ) & ( for z being Real
for A being Element of CosetSet M
for f being PartFunc of X,REAL st f in A holds
f2 . (z,A) = a.e-eq-class ((z (#) f),M) ) implies f1 = f2 )
assume that
A9: for z being Real
for A being Element of CosetSet M
for a being PartFunc of X,REAL st a in A holds
f1 . (z,A) = a.e-eq-class ((z (#) a),M) and
A10: for z being Real
for A being Element of CosetSet M
for a being PartFunc of X,REAL st a in A holds
f2 . (z,A) = a.e-eq-class ((z (#) a),M) ; :: thesis: f1 = f2

now :: thesis: for z being Element of REAL
for A being Element of CosetSet M holds f1 . (z,A) = f2 . (z,A)

let z be Element of REAL ; :: thesis: for A being Element of CosetSet M holds f1 . (z,A) = f2 . (z,A)
let A be Element of CosetSet M; :: thesis: f1 . (z,A) = f2 . (z,A)
A in CosetSet M ;
then consider a1 being PartFunc of X,REAL such that
A11: ( A = a.e-eq-class (a1,M) & a1 in L1_Functions M ) ;
thus f1 . (z,A) = a.e-eq-class ((z (#) a1),M) by A9, A11, Th38
.= f2 . (z,A) by A10, A11, Th38 ; :: thesis: verum

end;

hence f1 = f2 ; :: thesis: verum