let a, b, c, d, e be real number ; :: thesis: 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 ; :: thesis: ( 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'] ) ; :: thesis: integral (e (#) f),c,d = e * (integral f,c,d)
['a,b'] = [.a,b.] by A1, INTEGRA5:def 4;
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; :: thesis: verum