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 4;
hence integral ((c (#) f),a,b) = c * (integral (f,a,b)) by A1, A3, Th9; :: thesis: verum