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 A1: ( 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'] ) ; :: thesis: integral (e (#) f),c,d = e * (integral f,c,d)
then ['a,b'] = [.a,b.] by INTEGRA5:def 4;
then ( a <= c & d <= b ) by A1, XXREAL_1:1;
then ( f is_integrable_on ['c,d'] & f | ['c,d'] is bounded & ['c,d'] c= dom f ) by A1, Th18;
hence integral (e (#) f),c,d = e * (integral f,c,d) by A1, Th10; :: thesis: verum