let f, g be PartFunc of REAL,REAL; :: thesis: for a, b being Real st a < b & ['a,b'] c= dom f & ['a,b'] c= dom g & f is_left_ext_Riemann_integrable_on a,b & g is_left_ext_Riemann_integrable_on a,b holds

( f + g is_left_ext_Riemann_integrable_on a,b & ext_left_integral ((f + g),a,b) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) )

let a, b be Real; :: thesis: ( a < b & ['a,b'] c= dom f & ['a,b'] c= dom g & f is_left_ext_Riemann_integrable_on a,b & g is_left_ext_Riemann_integrable_on a,b implies ( f + g is_left_ext_Riemann_integrable_on a,b & ext_left_integral ((f + g),a,b) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) ) )

assume that

A1: a < b and

A2: ( ['a,b'] c= dom f & ['a,b'] c= dom g ) and

A3: f is_left_ext_Riemann_integrable_on a,b and

A4: g is_left_ext_Riemann_integrable_on a,b ; :: thesis: ( f + g is_left_ext_Riemann_integrable_on a,b & ext_left_integral ((f + g),a,b) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) )

consider Intg being PartFunc of REAL,REAL such that

A5: dom Intg = ].a,b.] and

A6: for x being Real st x in dom Intg holds

Intg . x = integral (g,x,b) and

A7: Intg is_right_convergent_in a and

A8: ext_left_integral (g,a,b) = lim_right (Intg,a) by A4, Def4;

consider Intf being PartFunc of REAL,REAL such that

A9: dom Intf = ].a,b.] and

A10: for x being Real st x in dom Intf holds

Intf . x = integral (f,x,b) and

A11: Intf is_right_convergent_in a and

A12: ext_left_integral (f,a,b) = lim_right (Intf,a) by A3, Def4;

set Intfg = Intf + Intg;

A13: ( dom (Intf + Intg) = ].a,b.] & ( for x being Real st x in dom (Intf + Intg) holds

(Intf + Intg) . x = integral ((f + g),x,b) ) )

ex g being Real st

( g < r & a < g & g in dom (Intf + Intg) )

for d being Real st a < d & d <= b holds

( f + g is_integrable_on ['d,b'] & (f + g) | ['d,b'] is bounded )

lim_right ((Intf + Intg),a) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) by A11, A12, A7, A8, A23, LIMFUNC2:54;

hence ext_left_integral ((f + g),a,b) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) by A13, A30, A36, Def4; :: thesis: verum

( f + g is_left_ext_Riemann_integrable_on a,b & ext_left_integral ((f + g),a,b) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) )

let a, b be Real; :: thesis: ( a < b & ['a,b'] c= dom f & ['a,b'] c= dom g & f is_left_ext_Riemann_integrable_on a,b & g is_left_ext_Riemann_integrable_on a,b implies ( f + g is_left_ext_Riemann_integrable_on a,b & ext_left_integral ((f + g),a,b) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) ) )

assume that

A1: a < b and

A2: ( ['a,b'] c= dom f & ['a,b'] c= dom g ) and

A3: f is_left_ext_Riemann_integrable_on a,b and

A4: g is_left_ext_Riemann_integrable_on a,b ; :: thesis: ( f + g is_left_ext_Riemann_integrable_on a,b & ext_left_integral ((f + g),a,b) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) )

consider Intg being PartFunc of REAL,REAL such that

A5: dom Intg = ].a,b.] and

A6: for x being Real st x in dom Intg holds

Intg . x = integral (g,x,b) and

A7: Intg is_right_convergent_in a and

A8: ext_left_integral (g,a,b) = lim_right (Intg,a) by A4, Def4;

consider Intf being PartFunc of REAL,REAL such that

A9: dom Intf = ].a,b.] and

A10: for x being Real st x in dom Intf holds

Intf . x = integral (f,x,b) and

A11: Intf is_right_convergent_in a and

A12: ext_left_integral (f,a,b) = lim_right (Intf,a) by A3, Def4;

set Intfg = Intf + Intg;

A13: ( dom (Intf + Intg) = ].a,b.] & ( for x being Real st x in dom (Intf + Intg) holds

(Intf + Intg) . x = integral ((f + g),x,b) ) )

proof

A23:
for r being Real st a < r holds
A14:
['a,b'] = [.a,b.]
by A1, INTEGRA5:def 3;

thus A15: dom (Intf + Intg) = (dom Intf) /\ (dom Intg) by VALUED_1:def 1

.= ].a,b.] by A9, A5 ; :: thesis: for x being Real st x in dom (Intf + Intg) holds

(Intf + Intg) . x = integral ((f + g),x,b)

let x be Real; :: thesis: ( x in dom (Intf + Intg) implies (Intf + Intg) . x = integral ((f + g),x,b) )

assume A16: x in dom (Intf + Intg) ; :: thesis: (Intf + Intg) . x = integral ((f + g),x,b)

then A17: a < x by A15, XXREAL_1:2;

then A18: [.x,b.] c= [.a,b.] by XXREAL_1:34;

A19: x <= b by A15, A16, XXREAL_1:2;

then A20: ( f is_integrable_on ['x,b'] & f | ['x,b'] is bounded ) by A3, A17;

['x,b'] = [.x,b.] by A19, INTEGRA5:def 3;

then A21: ( ['x,b'] c= dom f & ['x,b'] c= dom g ) by A2, A14, A18;

A22: ( g is_integrable_on ['x,b'] & g | ['x,b'] is bounded ) by A4, A17, A19;

thus (Intf + Intg) . x = (Intf . x) + (Intg . x) by A16, VALUED_1:def 1

.= (integral (f,x,b)) + (Intg . x) by A9, A10, A15, A16

.= (integral (f,x,b)) + (integral (g,x,b)) by A5, A6, A15, A16

.= integral ((f + g),x,b) by A19, A21, A20, A22, INTEGRA6:12 ; :: thesis: verum

end;thus A15: dom (Intf + Intg) = (dom Intf) /\ (dom Intg) by VALUED_1:def 1

.= ].a,b.] by A9, A5 ; :: thesis: for x being Real st x in dom (Intf + Intg) holds

(Intf + Intg) . x = integral ((f + g),x,b)

let x be Real; :: thesis: ( x in dom (Intf + Intg) implies (Intf + Intg) . x = integral ((f + g),x,b) )

assume A16: x in dom (Intf + Intg) ; :: thesis: (Intf + Intg) . x = integral ((f + g),x,b)

then A17: a < x by A15, XXREAL_1:2;

then A18: [.x,b.] c= [.a,b.] by XXREAL_1:34;

A19: x <= b by A15, A16, XXREAL_1:2;

then A20: ( f is_integrable_on ['x,b'] & f | ['x,b'] is bounded ) by A3, A17;

['x,b'] = [.x,b.] by A19, INTEGRA5:def 3;

then A21: ( ['x,b'] c= dom f & ['x,b'] c= dom g ) by A2, A14, A18;

A22: ( g is_integrable_on ['x,b'] & g | ['x,b'] is bounded ) by A4, A17, A19;

thus (Intf + Intg) . x = (Intf . x) + (Intg . x) by A16, VALUED_1:def 1

.= (integral (f,x,b)) + (Intg . x) by A9, A10, A15, A16

.= (integral (f,x,b)) + (integral (g,x,b)) by A5, A6, A15, A16

.= integral ((f + g),x,b) by A19, A21, A20, A22, INTEGRA6:12 ; :: thesis: verum

ex g being Real st

( g < r & a < g & g in dom (Intf + Intg) )

proof

then A30:
Intf + Intg is_right_convergent_in a
by A11, A7, LIMFUNC2:54;
let r be Real; :: thesis: ( a < r implies ex g being Real st

( g < r & a < g & g in dom (Intf + Intg) ) )

assume A24: a < r ; :: thesis: ex g being Real st

( g < r & a < g & g in dom (Intf + Intg) )

end;( g < r & a < g & g in dom (Intf + Intg) ) )

assume A24: a < r ; :: thesis: ex g being Real st

( g < r & a < g & g in dom (Intf + Intg) )

per cases
( b < r or not b < r )
;

end;

suppose A25:
b < r
; :: thesis: ex g being Real st

( g < r & a < g & g in dom (Intf + Intg) )

( g < r & a < g & g in dom (Intf + Intg) )

reconsider g = b as Real ;

take g ; :: thesis: ( g < r & a < g & g in dom (Intf + Intg) )

thus ( g < r & a < g & g in dom (Intf + Intg) ) by A1, A13, A25, XXREAL_1:2; :: thesis: verum

end;take g ; :: thesis: ( g < r & a < g & g in dom (Intf + Intg) )

thus ( g < r & a < g & g in dom (Intf + Intg) ) by A1, A13, A25, XXREAL_1:2; :: thesis: verum

suppose A26:
not b < r
; :: thesis: ex g being Real st

( g < r & a < g & g in dom (Intf + Intg) )

( g < r & a < g & g in dom (Intf + Intg) )

reconsider g = r - ((r - a) / 2) as Real ;

take g ; :: thesis: ( g < r & a < g & g in dom (Intf + Intg) )

A27: 0 < r - a by A24, XREAL_1:50;

then (r - a) / 2 < r - a by XREAL_1:216;

then A28: ((r - a) / 2) + (a - ((r - a) / 2)) < (r - a) + (a - ((r - a) / 2)) by XREAL_1:8;

A29: 0 < (r - a) / 2 by A27, XREAL_1:215;

then r - ((r - a) / 2) < b - 0 by A26, XREAL_1:15;

hence ( g < r & a < g & g in dom (Intf + Intg) ) by A13, A29, A28, XREAL_1:44, XXREAL_1:2; :: thesis: verum

end;take g ; :: thesis: ( g < r & a < g & g in dom (Intf + Intg) )

A27: 0 < r - a by A24, XREAL_1:50;

then (r - a) / 2 < r - a by XREAL_1:216;

then A28: ((r - a) / 2) + (a - ((r - a) / 2)) < (r - a) + (a - ((r - a) / 2)) by XREAL_1:8;

A29: 0 < (r - a) / 2 by A27, XREAL_1:215;

then r - ((r - a) / 2) < b - 0 by A26, XREAL_1:15;

hence ( g < r & a < g & g in dom (Intf + Intg) ) by A13, A29, A28, XREAL_1:44, XXREAL_1:2; :: thesis: verum

for d being Real st a < d & d <= b holds

( f + g is_integrable_on ['d,b'] & (f + g) | ['d,b'] is bounded )

proof

hence A36:
f + g is_left_ext_Riemann_integrable_on a,b
by A13, A30; :: thesis: ext_left_integral ((f + g),a,b) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b))
let d be Real; :: thesis: ( a < d & d <= b implies ( f + g is_integrable_on ['d,b'] & (f + g) | ['d,b'] is bounded ) )

assume A31: ( a < d & d <= b ) ; :: thesis: ( f + g is_integrable_on ['d,b'] & (f + g) | ['d,b'] is bounded )

then A32: ( ['d,b'] = [.d,b.] & [.d,b.] c= [.a,b.] ) by INTEGRA5:def 3, XXREAL_1:34;

['a,b'] = [.a,b.] by A1, INTEGRA5:def 3;

then A33: ( ['d,b'] c= dom f & ['d,b'] c= dom g ) by A2, A32;

A34: ( f is_integrable_on ['d,b'] & g is_integrable_on ['d,b'] ) by A3, A4, A31;

A35: ( f | ['d,b'] is bounded & g | ['d,b'] is bounded ) by A3, A4, A31;

then (f + g) | (['d,b'] /\ ['d,b']) is bounded by RFUNCT_1:83;

hence ( f + g is_integrable_on ['d,b'] & (f + g) | ['d,b'] is bounded ) by A33, A34, A35, INTEGRA6:11; :: thesis: verum

end;assume A31: ( a < d & d <= b ) ; :: thesis: ( f + g is_integrable_on ['d,b'] & (f + g) | ['d,b'] is bounded )

then A32: ( ['d,b'] = [.d,b.] & [.d,b.] c= [.a,b.] ) by INTEGRA5:def 3, XXREAL_1:34;

['a,b'] = [.a,b.] by A1, INTEGRA5:def 3;

then A33: ( ['d,b'] c= dom f & ['d,b'] c= dom g ) by A2, A32;

A34: ( f is_integrable_on ['d,b'] & g is_integrable_on ['d,b'] ) by A3, A4, A31;

A35: ( f | ['d,b'] is bounded & g | ['d,b'] is bounded ) by A3, A4, A31;

then (f + g) | (['d,b'] /\ ['d,b']) is bounded by RFUNCT_1:83;

hence ( f + g is_integrable_on ['d,b'] & (f + g) | ['d,b'] is bounded ) by A33, A34, A35, INTEGRA6:11; :: thesis: verum

lim_right ((Intf + Intg),a) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) by A11, A12, A7, A8, A23, LIMFUNC2:54;

hence ext_left_integral ((f + g),a,b) = (ext_left_integral (f,a,b)) + (ext_left_integral (g,a,b)) by A13, A30, A36, Def4; :: thesis: verum