let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,ExtREAL st f is_simple_func_in S & f is nonnegative holds
ex F being Finite_Sep_Sequence of S ex a being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & a . 1 = 0. & ( for n being Nat st 2 <= n & n in dom a holds
( 0. < a . n & a . n < +infty ) ) )
let S be SigmaField of X; :: thesis: for f being PartFunc of X,ExtREAL st f is_simple_func_in S & f is nonnegative holds
ex F being Finite_Sep_Sequence of S ex a being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & a . 1 = 0. & ( for n being Nat st 2 <= n & n in dom a holds
( 0. < a . n & a . n < +infty ) ) )
let f be PartFunc of X,ExtREAL; :: thesis: ( f is_simple_func_in S & f is nonnegative implies ex F being Finite_Sep_Sequence of S ex a being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & a . 1 = 0. & ( for n being Nat st 2 <= n & n in dom a holds
( 0. < a . n & a . n < +infty ) ) ) )
assume A1: ( f is_simple_func_in S & f is nonnegative ) ; :: thesis: ex F being Finite_Sep_Sequence of S ex a being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & a . 1 = 0. & ( for n being Nat st 2 <= n & n in dom a holds
( 0. < a . n & a . n < +infty ) ) )

per cases ( ex x being object st
( x in dom f & 0. = f . x ) or for x being object st x in dom f holds
0. <> f . x ) ;

suppose ex x being object st
( x in dom f & 0. = f . x ) ; :: thesis: ex F being Finite_Sep_Sequence of S ex a being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & a . 1 = 0. & ( for n being Nat st 2 <= n & n in dom a holds
( 0. < a . n & a . n < +infty ) ) )

hence ex F being Finite_Sep_Sequence of S ex a being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & a . 1 = 0. & ( for n being Nat st 2 <= n & n in dom a holds
( 0. < a . n & a . n < +infty ) ) ) by A1, Lm3; :: thesis: verum

end;

suppose for x being object st x in dom f holds
0. <> f . x ; :: thesis: ex F being Finite_Sep_Sequence of S ex a being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & a . 1 = 0. & ( for n being Nat st 2 <= n & n in dom a holds
( 0. < a . n & a . n < +infty ) ) )

hence ex F being Finite_Sep_Sequence of S ex a being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & a . 1 = 0. & ( for n being Nat st 2 <= n & n in dom a holds
( 0. < a . n & a . n < +infty ) ) ) by A1, Lm2; :: thesis: verum

end;

end;