let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,COMPLEX st ex A being Element of S st
( A = dom f & f is A -measurable ) & f is_integrable_on M holds
|.(Integral (M,f)).| <= Integral (M,|.f.|)
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,COMPLEX st ex A being Element of S st
( A = dom f & f is A -measurable ) & f is_integrable_on M holds
|.(Integral (M,f)).| <= Integral (M,|.f.|)
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,COMPLEX st ex A being Element of S st
( A = dom f & f is A -measurable ) & f is_integrable_on M holds
|.(Integral (M,f)).| <= Integral (M,|.f.|)
let f be PartFunc of X,COMPLEX; :: thesis: ( ex A being Element of S st
( A = dom f & f is A -measurable ) & f is_integrable_on M implies |.(Integral (M,f)).| <= Integral (M,|.f.|) )
assume that
A1: ex A being Element of S st
( A = dom f & f is A -measurable ) and
A2: f is_integrable_on M ; :: thesis: |.(Integral (M,f)).| <= Integral (M,|.f.|)

per cases ( Integral (M,f) = 0 or Integral (M,f) <> 0 ) ;

suppose Integral (M,f) = 0 ; :: thesis: |.(Integral (M,f)).| <= Integral (M,|.f.|)

hence |.(Integral (M,f)).| <= Integral (M,|.f.|) by A1, A2, Lm2; :: thesis: verum

end;

suppose Integral (M,f) <> 0 ; :: thesis: |.(Integral (M,f)).| <= Integral (M,|.f.|)

hence |.(Integral (M,f)).| <= Integral (M,|.f.|) by A1, A2, Lm3; :: thesis: verum

end;

end;