let X be non empty set ; :: thesis: for Y being set
for f being PartFunc of X, ExtREAL
for r being Real holds (r (#) f) | Y = r (#) (f | Y)
let Y be set ; :: thesis: for f being PartFunc of X, ExtREAL
for r being Real holds (r (#) f) | Y = r (#) (f | Y)
let f be PartFunc of X, ExtREAL ; :: thesis: for r being Real holds (r (#) f) | Y = r (#) (f | Y)
let r be Real; :: thesis: (r (#) f) | Y = r (#) (f | Y)
A1: dom ((r (#) f) | Y) = (dom (r (#) f)) /\ Y by RELAT_1:90
.= (dom f) /\ Y by MESFUNC1:def 6
.= dom (f | Y) by RELAT_1:90
.= dom (r (#) (f | Y)) by MESFUNC1:def 6 ;

now

let x be Element of X; :: thesis: ( x in dom ((r (#) f) | Y) implies ((r (#) f) | Y) . x = (r (#) (f | Y)) . x )
assume A2: x in dom ((r (#) f) | Y) ; :: thesis: ((r (#) f) | Y) . x = (r (#) (f | Y)) . x
then x in (dom (r (#) f)) /\ Y by RELAT_1:90;
then A3: ( x in dom (r (#) f) & x in Y ) by XBOOLE_0:def 4;
then x in dom f by MESFUNC1:def 6;
then x in (dom f) /\ Y by A3, XBOOLE_0:def 4;
then A4: x in dom (f | Y) by RELAT_1:90;
then A5: x in dom (r (#) (f | Y)) by MESFUNC1:def 6;
thus ((r (#) f) | Y) . x = (r (#) f) . x by A2, FUNCT_1:68
.= (R_EAL r) * (f . x) by A3, MESFUNC1:def 6
.= (R_EAL r) * ((f | Y) . x) by A4, FUNCT_1:68
.= (r (#) (f | Y)) . x by A5, MESFUNC1:def 6 ; :: thesis: verum

end;

hence (r (#) f) | Y = r (#) (f | Y) by A1, PARTFUN1:34; :: thesis: verum