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,ExtREAL
for c being Real st f is_integrable_on M holds
( c (#) f is_integrable_on M & Integral (M,(c (#) f)) = c * (Integral (M,f)) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for c being Real st f is_integrable_on M holds
( c (#) f is_integrable_on M & Integral (M,(c (#) f)) = c * (Integral (M,f)) )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL
for c being Real st f is_integrable_on M holds
( c (#) f is_integrable_on M & Integral (M,(c (#) f)) = c * (Integral (M,f)) )
let f be PartFunc of X,ExtREAL; :: thesis: for c being Real st f is_integrable_on M holds
( c (#) f is_integrable_on M & Integral (M,(c (#) f)) = c * (Integral (M,f)) )
let c be Real; :: thesis: ( f is_integrable_on M implies ( c (#) f is_integrable_on M & Integral (M,(c (#) f)) = c * (Integral (M,f)) ) )
assume A1: f is_integrable_on M ; :: thesis: ( c (#) f is_integrable_on M & Integral (M,(c (#) f)) = c * (Integral (M,f)) )
A2: integral+ (M,(max+ f)) <> +infty by A1;
consider A being Element of S such that
A3: A = dom f and
A4: f is A -measurable by A1;
A5: c (#) f is A -measurable by A3, A4, Th49;
A6: dom (max- f) = A by A3, MESFUNC2:def 3;
A7: integral+ (M,(max- f)) <> +infty by A1;
0 <= integral+ (M,(max- f)) by A1, Th96;
then reconsider I2 = integral+ (M,(max- f)) as Element of REAL by A7, XXREAL_0:14;
A8: max- f is nonnegative by Lm1;
0 <= integral+ (M,(max+ f)) by A1, Th96;
then reconsider I1 = integral+ (M,(max+ f)) as Element of REAL by A2, XXREAL_0:14;
A9: max+ f is nonnegative by Lm1;
A10: dom (c (#) f) = A by A3, MESFUNC1:def 6;
A11: dom (max+ f) = A by A3, MESFUNC2:def 2;