let a, b, c be real number ; :: thesis: for f being PartFunc of REAL ,REAL st a <= b & ['a,b'] c= dom f & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded holds
integral (c (#) f),a,b = c * (integral f,a,b)
let f be PartFunc of REAL ,REAL ; :: thesis: ( a <= b & ['a,b'] c= dom f & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded implies integral (c (#) f),a,b = c * (integral f,a,b) )
A1: c is Real by XREAL_0:def 1;
assume that
A2: a <= b and
A3: ( ['a,b'] c= dom f & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded ) ; :: thesis: integral (c (#) f),a,b = c * (integral f,a,b)
( integral f,a,b = integral f,['a,b'] & integral (c (#) f),a,b = integral (c (#) f),['a,b'] ) by A2, INTEGRA5:def 5;
hence integral (c (#) f),a,b = c * (integral f,a,b) by A1, A3, Th9; :: thesis: verum