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_measurable_on A ) & 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_measurable_on A ) & 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_measurable_on A ) & 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_measurable_on A ) & 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_measurable_on A ) 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;