let a, b, c, d, e be Real; :: 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 :: thesis: ( not c <= d implies integral ((e (#) f),c,d) = e * (integral (f,c,d)) )

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 4;
thus integral ((e (#) f),c,d) = - (integral ((e (#) f),['d,c'])) by A2, INTEGRA5:def 4
.= - (integral ((e (#) f),d,c)) by A2, INTEGRA5:def 4
.= - (e * (integral (f,d,c))) by A1, A2, Lm12
.= e * (- (integral (f,d,c)))
.= e * (integral (f,c,d)) by A2, A3, INTEGRA5:def 4 ; :: thesis: verum

end;

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