let a, b, c, d, e be Real; 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; ( 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'] )
; integral ((e (#) f),c,d) = e * (integral (f,c,d))
now ( not c <= d implies integral ((e (#) f),c,d) = e * (integral (f,c,d)) )assume A2:
not
c <= d
;
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
;
verum end;
hence
integral ((e (#) f),c,d) = e * (integral (f,c,d))
by A1, Lm12; verum