set C = CosetSet M;
let f1, f2 be Function of [:REAL,(CosetSet M):],(CosetSet M); ( ( 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)
; f1 = f2
now 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 ;
for A being Element of CosetSet M holds f1 . (z,A) = f2 . (z,A)let A be
Element of
CosetSet M;
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
;
verum end;
hence
f1 = f2
; verum