let f be PartFunc of REAL,REAL; :: thesis: for a, b being Real st a < b & ['a,b'] c= dom f & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded holds
ext_left_integral (f,a,b) = integral (f,a,b)

let a, b be Real; :: thesis: ( a < b & ['a,b'] c= dom f & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded implies ext_left_integral (f,a,b) = integral (f,a,b) )
assume that
A1: a < b and
A2: ['a,b'] c= dom f and
A3: f is_integrable_on ['a,b'] and
A4: f | ['a,b'] is bounded ; :: thesis: ext_left_integral (f,a,b) = integral (f,a,b)
reconsider AB = ].a,b.] as non empty Subset of REAL by ;
deffunc H1( Element of AB) -> Element of REAL = In ((integral (f,\$1,b)),REAL);
consider Intf being Function of AB,REAL such that
A5: for x being Element of AB holds Intf . x = H1(x) from A6: dom Intf = AB by FUNCT_2:def 1;
then reconsider Intf = Intf as PartFunc of REAL,REAL by RELSET_1:5;
consider M0 being Real such that
A7: for x being object st x in ['a,b'] /\ (dom f) holds
|.(f . x).| <= M0 by ;
reconsider M = M0 + 1 as Real ;
A8: for x being Real st x in ['a,b'] holds
|.(f . x).| < M
proof
A9: ['a,b'] /\ (dom f) = ['a,b'] by ;
let x be Real; :: thesis: ( x in ['a,b'] implies |.(f . x).| < M )
assume x in ['a,b'] ; :: thesis: |.(f . x).| < M
hence |.(f . x).| < M by ; :: thesis: verum
end;
a in { r where r is Real : ( a <= r & r <= b ) } by A1;
then a in [.a,b.] by RCOMP_1:def 1;
then a in ['a,b'] by ;
then |.(f . a).| < M by A8;
then A10: 0 < M by COMPLEX1:46;
A11: for g1 being Real st 0 < g1 holds
ex r being Real st
( a < r & ( for r1 being Real st r1 < r & a < r1 & r1 in dom Intf holds
|.((Intf . r1) - (integral (f,a,b))).| < g1 ) )
proof
let g1 be Real; :: thesis: ( 0 < g1 implies ex r being Real st
( a < r & ( for r1 being Real st r1 < r & a < r1 & r1 in dom Intf holds
|.((Intf . r1) - (integral (f,a,b))).| < g1 ) ) )

assume 0 < g1 ; :: thesis: ex r being Real st
( a < r & ( for r1 being Real st r1 < r & a < r1 & r1 in dom Intf holds
|.((Intf . r1) - (integral (f,a,b))).| < g1 ) )

then consider r being Real such that
A12: a < r and
A13: r < b and
A14: (r - a) * M < g1 by A1, A10, Th2;
take r ; :: thesis: ( a < r & ( for r1 being Real st r1 < r & a < r1 & r1 in dom Intf holds
|.((Intf . r1) - (integral (f,a,b))).| < g1 ) )

thus a < r by A12; :: thesis: for r1 being Real st r1 < r & a < r1 & r1 in dom Intf holds
|.((Intf . r1) - (integral (f,a,b))).| < g1

now :: thesis: for r1 being Real st a < r1 & r1 < r & r1 in dom Intf holds
|.((Intf . r1) - (integral (f,a,b))).| < g1
let r1 be Real; :: thesis: ( a < r1 & r1 < r & r1 in dom Intf implies |.((Intf . r1) - (integral (f,a,b))).| < g1 )
assume that
A15: a < r1 and
A16: r1 < r and
r1 in dom Intf ; :: thesis: |.((Intf . r1) - (integral (f,a,b))).| < g1
r1 < b by ;
then reconsider rr = r1 as Element of AB by ;
A17: Intf . r1 = H1(rr) by A5;
r1 - a < r - a by ;
then A18: M * (r1 - a) < M * (r - a) by ;
A19: ['a,b'] = [.a,b.] by ;
A20: r1 <= b by ;
then A21: r1 in ['a,b'] by ;
[.a,r1.] = ['a,r1'] by ;
then ['a,r1'] c= ['a,b'] by ;
then A22: for x being Real st x in ['a,r1'] holds
|.(f . x).| <= M by A8;
a in ['a,b'] by ;
then |.(integral (f,a,r1)).| <= M * (r1 - a) by A1, A2, A3, A4, A15, A21, A22, INTEGRA6:23;
then A23: |.(integral (f,a,r1)).| < M * (r - a) by ;
|.((Intf . r1) - (integral (f,a,b))).| = |.((integral (f,a,b)) - (Intf . r1)).| by COMPLEX1:60
.= |.(((integral (f,a,r1)) + (integral (f,r1,b))) - (integral (f,r1,b))).| by
.= |.(integral (f,a,r1)).| ;
hence |.((Intf . r1) - (integral (f,a,b))).| < g1 by ; :: thesis: verum
end;
hence for r1 being Real st r1 < r & a < r1 & r1 in dom Intf holds
|.((Intf . r1) - (integral (f,a,b))).| < g1 ; :: thesis: verum
end;
A24: for x being Real st x in dom Intf holds
Intf . x = integral (f,x,b)
proof
let x be Real; :: thesis: ( x in dom Intf implies Intf . x = integral (f,x,b) )
assume A25: x in dom Intf ; :: thesis: Intf . x = integral (f,x,b)
dom Intf = AB by FUNCT_2:def 1;
then reconsider x = x as Element of AB by A25;
Intf . x = H1(x) by A5;
hence Intf . x = integral (f,x,b) ; :: thesis: verum
end;
for r being Real st a < r holds
ex g being Real st
( g < r & a < g & g in dom Intf )
proof
let r be Real; :: thesis: ( a < r implies ex g being Real st
( g < r & a < g & g in dom Intf ) )

assume A26: a < r ; :: thesis: ex g being Real st
( g < r & a < g & g in dom Intf )

per cases ( b < r or not b < r ) ;
suppose A27: b < r ; :: thesis: ex g being Real st
( g < r & a < g & g in dom Intf )

reconsider g = b as Real ;
take g ; :: thesis: ( g < r & a < g & g in dom Intf )
g in ].a,b.] by ;
hence ( g < r & a < g & g in dom Intf ) by ; :: thesis: verum
end;
suppose A28: not b < r ; :: thesis: ex g being Real st
( g < r & a < g & g in dom Intf )

reconsider g = a + ((r - a) / 2) as Real ;
take g ; :: thesis: ( g < r & a < g & g in dom Intf )
A29: 0 < r - a by ;
then A30: a < g by ;
(r - a) / 2 < r - a by ;
then A31: ((r - a) / 2) + a < (r - a) + a by XREAL_1:8;
then g < b by ;
hence ( g < r & a < g & g in dom Intf ) by ; :: thesis: verum
end;
end;
end;
then A32: Intf is_right_convergent_in a by ;
then A33: integral (f,a,b) = lim_right (Intf,a) by ;
for d being Real st a < d & d <= b holds
( f is_integrable_on ['d,b'] & f | ['d,b'] is bounded ) by ;
then f is_left_ext_Riemann_integrable_on a,b by A6, A24, A32;
hence ext_left_integral (f,a,b) = integral (f,a,b) by A6, A24, A32, A33, Def4; :: thesis: verum