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_right_ext_Riemann_integrable_on a,b & g is_right_ext_Riemann_integrable_on a,b holds
( f + g is_right_ext_Riemann_integrable_on a,b & ext_right_integral ((f + g),a,b) = (ext_right_integral (f,a,b)) + (ext_right_integral (g,a,b)) )

let a, b be Real; :: thesis: ( a < b & ['a,b'] c= dom f & ['a,b'] c= dom g & f is_right_ext_Riemann_integrable_on a,b & g is_right_ext_Riemann_integrable_on a,b implies ( f + g is_right_ext_Riemann_integrable_on a,b & ext_right_integral ((f + g),a,b) = (ext_right_integral (f,a,b)) + (ext_right_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_right_ext_Riemann_integrable_on a,b and
A4: g is_right_ext_Riemann_integrable_on a,b ; :: thesis: ( f + g is_right_ext_Riemann_integrable_on a,b & ext_right_integral ((f + g),a,b) = (ext_right_integral (f,a,b)) + (ext_right_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,a,x) and
A7: Intg is_left_convergent_in b and
A8: ext_right_integral (g,a,b) = lim_left (Intg,b) by ;
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,a,x) and
A11: Intf is_left_convergent_in b and
A12: ext_right_integral (f,a,b) = lim_left (Intf,b) by ;
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),a,x) ) )
proof
A14: ['a,b'] = [.a,b.] by ;
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),a,x)

let x be Real; :: thesis: ( x in dom (Intf + Intg) implies (Intf + Intg) . x = integral ((f + g),a,x) )
assume A16: x in dom (Intf + Intg) ; :: thesis: (Intf + Intg) . x = integral ((f + g),a,x)
then A17: x < b by ;
then A18: [.a,x.] c= [.a,b.] by XXREAL_1:34;
A19: a <= x by ;
then A20: ( f is_integrable_on ['a,x'] & f | ['a,x'] is bounded ) by ;
['a,x'] = [.a,x.] by ;
then A21: ( ['a,x'] c= dom f & ['a,x'] c= dom g ) by A2, A14, A18;
A22: ( g is_integrable_on ['a,x'] & g | ['a,x'] is bounded ) by A4, A19, A17;
thus (Intf + Intg) . x = (Intf . x) + (Intg . x) by
.= (integral (f,a,x)) + (Intg . x) by A9, A10, A15, A16
.= (integral (f,a,x)) + (integral (g,a,x)) by A5, A6, A15, A16
.= integral ((f + g),a,x) by ; :: thesis: verum
end;
A23: for r being Real st r < b holds
ex g being Real st
( r < g & g < b & g in dom (Intf + Intg) )
proof
let r be Real; :: thesis: ( r < b implies ex g being Real st
( r < g & g < b & g in dom (Intf + Intg) ) )

assume A24: r < b ; :: thesis: ex g being Real st
( r < g & g < b & g in dom (Intf + Intg) )

per cases ( r < a or not r < a ) ;
suppose A25: r < a ; :: thesis: ex g being Real st
( r < g & g < b & g in dom (Intf + Intg) )

reconsider g = a as Real ;
take g ; :: thesis: ( r < g & g < b & g in dom (Intf + Intg) )
thus ( r < g & g < b & g in dom (Intf + Intg) ) by ; :: thesis: verum
end;
suppose not r < a ; :: thesis: ex g being Real st
( r < g & g < b & g in dom (Intf + Intg) )

then A26: a - a <= r - a by XREAL_1:9;
reconsider g = r + ((b - r) / 2) as Real ;
take g ; :: thesis: ( r < g & g < b & g in dom (Intf + Intg) )
A27: 0 < b - r by ;
then (b - r) / 2 < b - r by XREAL_1:216;
then A28: ((b - r) / 2) + r < (b - r) + r by XREAL_1:8;
r < g by ;
then A29: r - (r - a) < g - 0 by ;
0 < (b - r) / 2 by ;
hence ( r < g & g < b & g in dom (Intf + Intg) ) by ; :: thesis: verum
end;
end;
end;
then A30: Intf + Intg is_left_convergent_in b by ;
for d being Real st a <= d & d < b holds
( f + g is_integrable_on ['a,d'] & (f + g) | ['a,d'] is bounded )
proof
let d be Real; :: thesis: ( a <= d & d < b implies ( f + g is_integrable_on ['a,d'] & (f + g) | ['a,d'] is bounded ) )
assume A31: ( a <= d & d < b ) ; :: thesis: ( f + g is_integrable_on ['a,d'] & (f + g) | ['a,d'] is bounded )
then A32: ( ['a,d'] = [.a,d.] & [.a,d.] c= [.a,b.] ) by ;
['a,b'] = [.a,b.] by ;
then A33: ( ['a,d'] c= dom f & ['a,d'] c= dom g ) by ;
A34: ( f is_integrable_on ['a,d'] & g is_integrable_on ['a,d'] ) by A3, A4, A31;
A35: ( f | ['a,d'] is bounded & g | ['a,d'] is bounded ) by A3, A4, A31;
then (f + g) | (['a,d'] /\ ['a,d']) is bounded by RFUNCT_1:83;
hence ( f + g is_integrable_on ['a,d'] & (f + g) | ['a,d'] is bounded ) by ; :: thesis: verum
end;
hence A36: f + g is_right_ext_Riemann_integrable_on a,b by ; :: thesis: ext_right_integral ((f + g),a,b) = (ext_right_integral (f,a,b)) + (ext_right_integral (g,a,b))
lim_left ((Intf + Intg),b) = (ext_right_integral (f,a,b)) + (ext_right_integral (g,a,b)) by ;
hence ext_right_integral ((f + g),a,b) = (ext_right_integral (f,a,b)) + (ext_right_integral (g,a,b)) by ; :: thesis: verum