let a, b, c, d be Real; for f, g being PartFunc of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b'] holds
( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )
let f, g be PartFunc of REAL,REAL; ( a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b'] implies ( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) ) )
assume A1:
( a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b'] )
; ( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )
now ( not c <= d implies ( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) ) )assume A2:
not
c <= d
;
( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )then A3:
integral (
f,
c,
d)
= - (integral (f,['d,c']))
by INTEGRA5:def 4;
A4:
integral (
g,
c,
d)
= - (integral (g,['d,c']))
by A2, INTEGRA5:def 4;
integral (
(f + g),
c,
d)
= - (integral ((f + g),['d,c']))
by A2, INTEGRA5:def 4;
hence integral (
(f + g),
c,
d) =
- (integral ((f + g),d,c))
by A2, INTEGRA5:def 4
.=
- ((integral (f,d,c)) + (integral (g,d,c)))
by A1, A2, Lm11
.=
(- (integral (f,d,c))) - (integral (g,d,c))
.=
(integral (f,c,d)) - (integral (g,d,c))
by A2, A3, INTEGRA5:def 4
.=
(integral (f,c,d)) + (integral (g,c,d))
by A2, A4, INTEGRA5:def 4
;
integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d))
integral (
(f - g),
c,
d)
= - (integral ((f - g),['d,c']))
by A2, INTEGRA5:def 4;
hence integral (
(f - g),
c,
d) =
- (integral ((f - g),d,c))
by A2, INTEGRA5:def 4
.=
- ((integral (f,d,c)) - (integral (g,d,c)))
by A1, A2, Lm11
.=
- ((integral (f,d,c)) + (integral (g,c,d)))
by A2, A4, INTEGRA5:def 4
.=
(- (integral (f,d,c))) - (integral (g,c,d))
.=
(integral (f,c,d)) - (integral (g,c,d))
by A2, A3, INTEGRA5:def 4
;
verum end;
hence
( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )
by A1, Lm11; verum