let a, b, c, d, e be real number ; :: thesis: for f being PartFunc of REAL ,REAL st a <= b & 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 & 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 & 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)

now

assume A2: not c <= d ; :: thesis: integral (e (#) f),c,d = e * (integral f,c,d)
then A3: integral f,c,d = - (integral f,['d,c']) by INTEGRA5:def 5;
thus integral (e (#) f),c,d = - (integral (e (#) f),['d,c']) by A2, INTEGRA5:def 5
.= - (integral (e (#) f),d,c) by A2, INTEGRA5:def 5
.= - (e * (integral f,d,c)) by A1, A2, Lm12
.= e * (- (integral f,d,c))
.= e * (integral f,c,d) by A2, A3, INTEGRA5:def 5 ; :: thesis: verum

end;

hence integral (e (#) f),c,d = e * (integral f,c,d) by A1, Lm12; :: thesis: verum