let a, b, c, d 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 (f,a,d) = (integral (f,a,c)) + (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 (f,a,d) = (integral (f,a,c)) + (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 (f,a,d) = (integral (f,a,c)) + (integral (f,c,d))
now ( not c <= d implies integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d)) )assume A2:
not
c <= d
;
integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d))then
integral (
f,
a,
c)
= (integral (f,a,d)) + (integral (f,d,c))
by A1, Lm4;
then A3:
integral (
f,
a,
d)
= (integral (f,a,c)) - (integral (f,d,c))
;
integral (
f,
c,
d)
= - (integral (f,['d,c']))
by A2, INTEGRA5:def 4;
hence
integral (
f,
a,
d)
= (integral (f,a,c)) + (integral (f,c,d))
by A2, A3, INTEGRA5:def 4;
verum end;
hence
integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d))
by A1, Lm4; verum