let X1, X2 be non empty set ; :: thesis: for S1 being SigmaField of X1
for S2 being SigmaField of X2
for f being PartFunc of [:X1,X2:],ExtREAL
for x being Element of X1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 (f,x) is Measurable-X-section (E,x) -measurable & ProjPMap2 (f,y) is Measurable-Y-section (E,y) -measurable )
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for f being PartFunc of [:X1,X2:],ExtREAL
for x being Element of X1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 (f,x) is Measurable-X-section (E,x) -measurable & ProjPMap2 (f,y) is Measurable-Y-section (E,y) -measurable )
let S2 be SigmaField of X2; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL
for x being Element of X1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 (f,x) is Measurable-X-section (E,x) -measurable & ProjPMap2 (f,y) is Measurable-Y-section (E,y) -measurable )
let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: for x being Element of X1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 (f,x) is Measurable-X-section (E,x) -measurable & ProjPMap2 (f,y) is Measurable-Y-section (E,y) -measurable )
let x be Element of X1; :: thesis: for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 (f,x) is Measurable-X-section (E,x) -measurable & ProjPMap2 (f,y) is Measurable-Y-section (E,y) -measurable )
let y be Element of X2; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 (f,x) is Measurable-X-section (E,x) -measurable & ProjPMap2 (f,y) is Measurable-Y-section (E,y) -measurable )
let A be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: ( A c= dom f & f is A -measurable implies ( ProjPMap1 (f,x) is Measurable-X-section (A,x) -measurable & ProjPMap2 (f,y) is Measurable-Y-section (A,y) -measurable ) )
assume that
A1: A c= dom f and
A2: f is A -measurable ; :: thesis: ( ProjPMap1 (f,x) is Measurable-X-section (A,x) -measurable & ProjPMap2 (f,y) is Measurable-Y-section (A,y) -measurable )
( X-section (A,x) c= X-section ((dom f),x) & Y-section (A,y) c= Y-section ((dom f),y) ) by A1, MEASUR11:20, MEASUR11:21;
then ( Measurable-X-section (A,x) c= X-section ((dom f),x) & Measurable-Y-section (A,y) c= Y-section ((dom f),y) ) by MEASUR11:def 6, MEASUR11:def 7;
then A3: ( Measurable-X-section (A,x) c= dom (ProjPMap1 (f,x)) & Measurable-Y-section (A,y) c= dom (ProjPMap2 (f,y)) ) by Def3, Def4;
( ProjPMap1 ((max+ f),x) is Measurable-X-section (A,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (A,y) -measurable & ProjPMap1 ((max- f),x) is Measurable-X-section (A,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (A,y) -measurable ) by A1, A2, Lm4;
then ( max+ (ProjPMap1 (f,x)) is Measurable-X-section (A,x) -measurable & max+ (ProjPMap2 (f,y)) is Measurable-Y-section (A,y) -measurable & max- (ProjPMap1 (f,x)) is Measurable-X-section (A,x) -measurable & max- (ProjPMap2 (f,y)) is Measurable-Y-section (A,y) -measurable ) by Th45, Th46;
hence ( ProjPMap1 (f,x) is Measurable-X-section (A,x) -measurable & ProjPMap2 (f,y) is Measurable-Y-section (A,y) -measurable ) by A3, MESFUN11:10; :: thesis: verum