let a, b, c, d, e be Real; for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
integral ((e (#) f),c,d) = e * (integral (f,c,d))
let f be PartFunc of REAL,REAL; ( a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] implies integral ((e (#) f),c,d) = e * (integral (f,c,d)) )
assume that
A1:
a <= b
and
A2:
c <= d
and
A3:
( f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f )
and
A4:
( c in ['a,b'] & d in ['a,b'] )
; integral ((e (#) f),c,d) = e * (integral (f,c,d))
['a,b'] = [.a,b.]
by A1, INTEGRA5:def 3;
then A5:
( a <= c & d <= b )
by A4, XXREAL_1:1;
then A6:
['c,d'] c= dom f
by A2, A3, Th18;
( f is_integrable_on ['c,d'] & f | ['c,d'] is bounded )
by A2, A3, A5, Th18;
hence
integral ((e (#) f),c,d) = e * (integral (f,c,d))
by A2, A6, Th10; verum