set V = { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M )
}
;
A1: { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M ) } c= PFuncs (X,REAL)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M )
}
or x in PFuncs (X,REAL) )

assume x in { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M )
}
; :: thesis: x in PFuncs (X,REAL)
then ex f being PartFunc of X,REAL st
( x = f & ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M ) ) ;
hence x in PFuncs (X,REAL) by PARTFUN1:45; :: thesis: verum
end;
reconsider g = X --> (In (0,REAL)) as Function of X,REAL by FUNCOP_1:46;
reconsider Ef = X as Element of S by MEASURE1:34;
set h = (abs g) to_power k;
A2: dom g = X by FUNCOP_1:13;
for x being set st x in dom g holds
g . x = 0 by FUNCOP_1:7;
then A3: g is Ef -measurable by ;
Ef ` = {} by XBOOLE_1:37;
then A4: M . (Ef `) = 0 by VALUED_0:def 19;
for x being object st x in dom (X --> 0) holds
0 <= (X --> 0) . x ;
then abs g = X --> 0 by ;
then A5: (abs g) to_power k = g by Th12;
then for x being Element of X st x in dom ((abs g) to_power k) holds
((abs g) to_power k) . x = 0 by FUNCOP_1:7;
then (abs g) to_power k is_integrable_on M by Th15, A5, A2;
then g in { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M )
}
by A3, A4, A2;
hence { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M ) } is non empty Subset of () by A1; :: thesis: verum