:: LPSPACE2 semantic presentation

begin


theorem Th1: :: LPSPACE2:1
for m, n being positive real number st (1 / m) + (1 / n) = 1 holds
m > 1
proof
let m, n be positive real number ; ::_thesis: ( (1 / m) + (1 / n) = 1 implies m > 1 )
assume  (1 / m) + (1 / n) = 1 ; ::_thesis: m > 1
then A1: 1 / n = 1 - (1 / m) ;
assume  m <= 1 ; ::_thesis: contradiction
then  1 <= 1 / m by XREAL_1:181;
hence  contradiction by A1, XREAL_1:47; ::_thesis: verum
end;

theorem Th2: :: LPSPACE2:2
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for A being Element of S
for f being PartFunc of X,ExtREAL st A = dom f & f is_measurable_on A & f is nonnegative holds
( Integral (M,f) in REAL iff f is_integrable_on M )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for A being Element of S
for f being PartFunc of X,ExtREAL st A = dom f & f is_measurable_on A & f is nonnegative holds
( Integral (M,f) in REAL iff f is_integrable_on M )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for A being Element of S
for f being PartFunc of X,ExtREAL st A = dom f & f is_measurable_on A & f is nonnegative holds
( Integral (M,f) in REAL iff f is_integrable_on M )
let M be sigma_Measure of S; ::_thesis: for A being Element of S
for f being PartFunc of X,ExtREAL st A = dom f & f is_measurable_on A & f is nonnegative holds
( Integral (M,f) in REAL iff f is_integrable_on M )
let A be Element of S; ::_thesis: for f being PartFunc of X,ExtREAL st A = dom f & f is_measurable_on A & f is nonnegative holds
( Integral (M,f) in REAL iff f is_integrable_on M )
let f be PartFunc of X,ExtREAL; ::_thesis: ( A = dom f & f is_measurable_on A & f is nonnegative implies ( Integral (M,f) in REAL iff f is_integrable_on M ) )
assume A1: ( A = dom f & f is_measurable_on A & f is nonnegative ) ; ::_thesis: ( Integral (M,f) in REAL iff f is_integrable_on M )
A2: now__::_thesis:_(_f_is_integrable_on_M_implies_Integral_(M,f)_in_REAL_)
assume  f is_integrable_on M ; ::_thesis: Integral (M,f) in REAL
then  ( -infty < Integral (M,f) & Integral (M,f) < +infty ) by MESFUNC5:96;
hence  Integral (M,f) in REAL by XXREAL_0:14; ::_thesis: verum
end;
now__::_thesis:_(_Integral_(M,f)_in_REAL_implies_f_is_integrable_on_M_)
assume A3: Integral (M,f) in REAL ; ::_thesis: f is_integrable_on M
A4: ( dom (max- f) = A & max- f is_measurable_on A ) by A1, MESFUNC2:26, MESFUNC2:def_3;
A5: ( dom (max+ f) = A & max+ f is_measurable_on A ) by A1, MESFUNC2:25, MESFUNC2:def_2;
 for x being Element of X holds 0 <= (max+ f) . x by MESFUNC2:12;
then  max+ f is nonnegative by SUPINF_2:39;
then A6: Integral (M,(max+ f)) = integral+ (M,(max+ f)) by A5, MESFUNC5:88;
A7: for x being Element of X st x in dom f holds
(max+ f) . x = f . x
proof
let x be Element of X; ::_thesis: ( x in dom f implies (max+ f) . x = f . x )
A8: f . x >= 0 by A1, SUPINF_2:39;
assume  x in dom f ; ::_thesis: (max+ f) . x = f . x
then  (max+ f) . x = max ((f . x),0) by A1, A5, MESFUNC2:def_2;
hence  (max+ f) . x = f . x by A8, XXREAL_0:def_10; ::_thesis: verum
end;
then  max+ f = f by A1, A5, PARTFUN1:5;
then A9: Integral (M,(max+ f)) < +infty by A3, XXREAL_0:9;
 for x being Element of X holds 0 <= (max- f) . x by MESFUNC2:13;
then  max- f is nonnegative by SUPINF_2:39;
then A10: Integral (M,(max- f)) = integral+ (M,(max- f)) by A4, MESFUNC5:88;
 for x being Element of X st x in dom (max- f) holds
0 = (max- f) . x
proof
let x be Element of X; ::_thesis: ( x in dom (max- f) implies 0 = (max- f) . x )
assume  x in dom (max- f) ; ::_thesis: 0 = (max- f) . x
 (max+ f) . x = f . x by A1, A5, A7, PARTFUN1:5;
hence  0 = (max- f) . x by MESFUNC2:19; ::_thesis: verum
end;
then  Integral (M,(max- f)) = 0 by A4, LPSPACE1:22;
hence  f is_integrable_on M by A1, A6, A9, A10, MESFUNC5:def_17; ::_thesis: verum
end;
hence  ( Integral (M,f) in REAL iff f is_integrable_on M ) by A2; ::_thesis: verum
end;

definition
let r be real number ;
attrr is geq_than_1  means :Def1: :: LPSPACE2:def 1
1 <= r;
end;

:: deftheorem Def1 defines geq_than_1 LPSPACE2:def_1_:_
 for r being real number holds
( r is geq_than_1 iff 1 <= r );

registration
cluster real geq_than_1  ->  positive real for set ;
coherence
 for b1 being real number st b1 is geq_than_1 holds
b1 is positive
proof
let r be real number ; ::_thesis: ( r is geq_than_1 implies r is positive )
assume  1 <= r ; :: according to LPSPACE2:def_1 ::_thesis: r is positive
hence  r is positive ; ::_thesis: verum
end;
end;

registration
cluster ext-real V38() real geq_than_1 for Element of REAL ;
existence
 ex b1 being Real st b1 is geq_than_1
proof
take  1 ; ::_thesis: 1 is geq_than_1
thus  1 is geq_than_1 by Def1; ::_thesis: verum
end;
end;

theorem Th3: :: LPSPACE2:3
for a, b, p being Real st 0 < p & 0 <= a & a < b holds
a to_power p < b to_power p
proof
let a, b, p be Real; ::_thesis: ( 0 < p & 0 <= a & a < b implies a to_power p < b to_power p )
assume A1: ( 0 < p & 0 <= a & a < b ) ; ::_thesis: a to_power p < b to_power p
now__::_thesis:_(_a_=_0_implies_a_to_power_p_<_b_to_power_p_)
assume  a = 0 ; ::_thesis: a to_power p < b to_power p
then  a to_power p = 0 by A1, POWER:def_2;
hence  a to_power p < b to_power p by A1, POWER:34; ::_thesis: verum
end;
hence  a to_power p < b to_power p by A1, POWER:37; ::_thesis: verum
end;

theorem Th4: :: LPSPACE2:4
for a, b being Real st a >= 0 & b > 0 holds
a to_power b >= 0
proof
let a, b be Real; ::_thesis: ( a >= 0 & b > 0 implies a to_power b >= 0 )
assume A1: a >= 0 ; ::_thesis: ( not b > 0 or a to_power b >= 0 )
assume  b > 0 ; ::_thesis: a to_power b >= 0
then  ( a = 0 implies a to_power b >= 0 ) by POWER:def_2;
hence  a to_power b >= 0 by A1, POWER:34; ::_thesis: verum
end;

theorem Th5: :: LPSPACE2:5
for a, b, c being Real st a >= 0 & b >= 0 & c > 0 holds
(a * b) to_power c = (a to_power c) * (b to_power c)
proof
let a, b, c be Real; ::_thesis: ( a >= 0 & b >= 0 & c > 0 implies (a * b) to_power c = (a to_power c) * (b to_power c) )
assume that
A1: ( a >= 0 & b >= 0 ) and
A2: c > 0 ; ::_thesis: (a * b) to_power c = (a to_power c) * (b to_power c)
now__::_thesis:_(_(_a_=_0_or_b_=_0_)_implies_(a_*_b)_to_power_c_=_(a_to_power_c)_*_(b_to_power_c)_)
assume A3: ( a = 0 or b = 0 ) ; ::_thesis: (a * b) to_power c = (a to_power c) * (b to_power c)
then  (a * b) to_power c = 0 by A2, POWER:def_2;
hence  (a * b) to_power c = (a to_power c) * (b to_power c) by A3; ::_thesis: verum
end;
hence  (a * b) to_power c = (a to_power c) * (b to_power c) by A1, POWER:30; ::_thesis: verum
end;

theorem Th6: :: LPSPACE2:6
for X being non empty set
for a, b being Real
for f being PartFunc of X,REAL st f is nonnegative & a > 0 & b > 0 holds
(f to_power a) to_power b = f to_power (a * b)
proof
let X be non empty set ; ::_thesis: for a, b being Real
for f being PartFunc of X,REAL st f is nonnegative & a > 0 & b > 0 holds
(f to_power a) to_power b = f to_power (a * b)
let a, b be Real; ::_thesis: for f being PartFunc of X,REAL st f is nonnegative & a > 0 & b > 0 holds
(f to_power a) to_power b = f to_power (a * b)
let f be PartFunc of X,REAL; ::_thesis: ( f is nonnegative & a > 0 & b > 0 implies (f to_power a) to_power b = f to_power (a * b) )
assume A1: ( f is nonnegative & a > 0 & b > 0 ) ; ::_thesis: (f to_power a) to_power b = f to_power (a * b)
A2: ( dom (f to_power a) = dom f & dom ((f to_power a) to_power b) = dom (f to_power a) & dom (f to_power (a * b)) = dom f ) by MESFUN6C:def_4;
 for x being set st x in dom ((f to_power a) to_power b) holds
((f to_power a) to_power b) . x = (f to_power (a * b)) . x
proof
let x be set ; ::_thesis: ( x in dom ((f to_power a) to_power b) implies ((f to_power a) to_power b) . x = (f to_power (a * b)) . x )
assume A3: x in dom ((f to_power a) to_power b) ; ::_thesis: ((f to_power a) to_power b) . x = (f to_power (a * b)) . x
then A4: ((f to_power a) to_power b) . x = ((f to_power a) . x) to_power b by MESFUN6C:def_4
.= ((f . x) to_power a) to_power b by A2, A3, MESFUN6C:def_4 ;
A5: (f to_power (a * b)) . x = (f . x) to_power (a * b) by A2, A3, MESFUN6C:def_4;
then A6: ( f . x > 0 implies ((f to_power a) to_power b) . x = (f to_power (a * b)) . x ) by A4, POWER:33;
now__::_thesis:_(_f_._x_=_0_implies_((f_to_power_a)_to_power_b)_._x_=_(f_to_power_(a_*_b))_._x_)
assume A7: f . x = 0 ; ::_thesis: ((f to_power a) to_power b) . x = (f to_power (a * b)) . x
then  ((f to_power a) to_power b) . x = 0 to_power b by A1, A4, POWER:def_2;
then  ((f to_power a) to_power b) . x = 0 by A1, POWER:def_2;
hence  ((f to_power a) to_power b) . x = (f to_power (a * b)) . x by A1, A7, A5, POWER:def_2; ::_thesis: verum
end;
hence  ((f to_power a) to_power b) . x = (f to_power (a * b)) . x by A6, A1, MESFUNC6:51; ::_thesis: verum
end;
hence  (f to_power a) to_power b = f to_power (a * b) by A2, FUNCT_1:2; ::_thesis: verum
end;

theorem Th7: :: LPSPACE2:7
for X being non empty set
for a, b being Real
for f being PartFunc of X,REAL st f is nonnegative & a > 0 & b > 0 holds
(f to_power a) (#) (f to_power b) = f to_power (a + b)
proof
let X be non empty set ; ::_thesis: for a, b being Real
for f being PartFunc of X,REAL st f is nonnegative & a > 0 & b > 0 holds
(f to_power a) (#) (f to_power b) = f to_power (a + b)
let a, b be Real; ::_thesis: for f being PartFunc of X,REAL st f is nonnegative & a > 0 & b > 0 holds
(f to_power a) (#) (f to_power b) = f to_power (a + b)
let f be PartFunc of X,REAL; ::_thesis: ( f is nonnegative & a > 0 & b > 0 implies (f to_power a) (#) (f to_power b) = f to_power (a + b) )
assume A1: ( f is nonnegative & a > 0 & b > 0 ) ; ::_thesis: (f to_power a) (#) (f to_power b) = f to_power (a + b)
A2: ( dom (f to_power a) = dom f & dom (f to_power b) = dom f ) by MESFUN6C:def_4;
A3: dom ((f to_power a) (#) (f to_power b)) = (dom (f to_power a)) /\ (dom (f to_power b)) by VALUED_1:def_4;
then A4: dom ((f to_power a) (#) (f to_power b)) = dom (f to_power (a + b)) by A2, MESFUN6C:def_4;
 for x being set st x in dom ((f to_power a) (#) (f to_power b)) holds
((f to_power a) (#) (f to_power b)) . x = (f to_power (a + b)) . x
proof
let x be set ; ::_thesis: ( x in dom ((f to_power a) (#) (f to_power b)) implies ((f to_power a) (#) (f to_power b)) . x = (f to_power (a + b)) . x )
assume A5: x in dom ((f to_power a) (#) (f to_power b)) ; ::_thesis: ((f to_power a) (#) (f to_power b)) . x = (f to_power (a + b)) . x
then  ( (f to_power a) . x = (f . x) to_power a & (f to_power b) . x = (f . x) to_power b ) by A2, A3, MESFUN6C:def_4;
then A6: ((f to_power a) (#) (f to_power b)) . x = ((f . x) to_power a) * ((f . x) to_power b) by A5, VALUED_1:def_4;
A7: (f to_power (a + b)) . x = (f . x) to_power (a + b) by A4, A5, MESFUN6C:def_4;
then A8: ( f . x > 0 implies ((f to_power a) (#) (f to_power b)) . x = (f to_power (a + b)) . x ) by A6, POWER:27;
now__::_thesis:_(_f_._x_=_0_implies_((f_to_power_a)_(#)_(f_to_power_b))_._x_=_(f_to_power_(a_+_b))_._x_)
assume A9: f . x = 0 ; ::_thesis: ((f to_power a) (#) (f to_power b)) . x = (f to_power (a + b)) . x
then  ((f to_power a) (#) (f to_power b)) . x = 0 * (0 to_power b) by A1, A6, POWER:def_2;
hence  ((f to_power a) (#) (f to_power b)) . x = (f to_power (a + b)) . x by A7, A1, A9, POWER:def_2; ::_thesis: verum
end;
hence  ((f to_power a) (#) (f to_power b)) . x = (f to_power (a + b)) . x by A1, A8, MESFUNC6:51; ::_thesis: verum
end;
hence  (f to_power a) (#) (f to_power b) = f to_power (a + b) by A4, FUNCT_1:2; ::_thesis: verum
end;

theorem Th8: :: LPSPACE2:8
for X being non empty set
for f being PartFunc of X,REAL holds f to_power 1 = f
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL holds f to_power 1 = f
let f be PartFunc of X,REAL; ::_thesis: f to_power 1 = f
A1: dom (f to_power 1) = dom f by MESFUN6C:def_4;
 for x being set st x in dom (f to_power 1) holds
(f to_power 1) . x = f . x
proof
let x be set ; ::_thesis: ( x in dom (f to_power 1) implies (f to_power 1) . x = f . x )
assume  x in dom (f to_power 1) ; ::_thesis: (f to_power 1) . x = f . x
then  (f to_power 1) . x = (f . x) to_power 1 by MESFUN6C:def_4;
hence  (f to_power 1) . x = f . x by POWER:25; ::_thesis: verum
end;
hence  f to_power 1 = f by A1, FUNCT_1:2; ::_thesis: verum
end;

theorem Th9: :: LPSPACE2:9
for seq1, seq2 being Real_Sequence
for k being positive Real st ( for n being Element of NAT holds
( seq1 . n = (seq2 . n) to_power k & seq2 . n >= 0 ) ) holds
( seq1 is convergent iff seq2 is convergent )
proof
let seq1, seq2 be Real_Sequence; ::_thesis: for k being positive Real st ( for n being Element of NAT holds
( seq1 . n = (seq2 . n) to_power k & seq2 . n >= 0 ) ) holds
( seq1 is convergent iff seq2 is convergent )
let k be positive Real; ::_thesis: ( ( for n being Element of NAT holds
( seq1 . n = (seq2 . n) to_power k & seq2 . n >= 0 ) ) implies ( seq1 is convergent iff seq2 is convergent ) )
assume A1: for n being Element of NAT holds
( seq1 . n = (seq2 . n) to_power k & seq2 . n >= 0 ) ; ::_thesis: ( seq1 is convergent iff seq2 is convergent )
A2: for n being Element of NAT holds seq1 . n >= 0
proof
let n be Element of NAT ; ::_thesis: seq1 . n >= 0
 (seq2 . n) to_power k >= 0 by A1, Th4;
hence  seq1 . n >= 0 by A1; ::_thesis: verum
end;
thus  ( seq1 is convergent implies seq2 is convergent ) ::_thesis: ( seq2 is convergent implies seq1 is convergent )
proof
assume A3: seq1 is convergent ; ::_thesis: seq2 is convergent
 for n being Element of NAT holds seq2 . n = (seq1 . n) to_power (1 / k)
proof
let n be Element of NAT ; ::_thesis: seq2 . n = (seq1 . n) to_power (1 / k)
(seq1 . n) to_power (1 / k) = ((seq2 . n) to_power k) to_power (1 / k) by A1
.= (seq2 . n) to_power (k * (1 / k)) by A1, HOLDER_1:2
.= (seq2 . n) to_power 1 by XCMPLX_1:106 ;
hence  seq2 . n = (seq1 . n) to_power (1 / k) by POWER:25; ::_thesis: verum
end;
hence  seq2 is convergent by A2, A3, HOLDER_1:10; ::_thesis: verum
end;
assume  seq2 is convergent ; ::_thesis: seq1 is convergent
hence  seq1 is convergent by A1, HOLDER_1:10; ::_thesis: verum
end;

theorem Th10: :: LPSPACE2:10
for seq being Real_Sequence
for n, m being Element of NAT st m <= n holds
( abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= ((Partial_Sums (abs seq)) . n) - ((Partial_Sums (abs seq)) . m) & abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= (Partial_Sums (abs seq)) . n )
proof
let seq be Real_Sequence; ::_thesis: for n, m being Element of NAT st m <= n holds
( abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= ((Partial_Sums (abs seq)) . n) - ((Partial_Sums (abs seq)) . m) & abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= (Partial_Sums (abs seq)) . n )
let n, m be Element of NAT ; ::_thesis: ( m <= n implies ( abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= ((Partial_Sums (abs seq)) . n) - ((Partial_Sums (abs seq)) . m) & abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= (Partial_Sums (abs seq)) . n ) )
assume A1: m <= n ; ::_thesis: ( abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= ((Partial_Sums (abs seq)) . n) - ((Partial_Sums (abs seq)) . m) & abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= (Partial_Sums (abs seq)) . n )
A2: for n being Element of NAT holds (abs seq) . n >= 0
proof
let n be Element of NAT ; ::_thesis: (abs seq) . n >= 0
 abs (seq . n) = (abs seq) . n by SEQ_1:12;
hence  (abs seq) . n >= 0 by COMPLEX1:46; ::_thesis: verum
end;
then A3: abs (((Partial_Sums (abs seq)) . n) - ((Partial_Sums (abs seq)) . m)) = ((Partial_Sums (abs seq)) . n) - ((Partial_Sums (abs seq)) . m) by A1, COMSEQ_3:9;
 (Partial_Sums (abs seq)) . m >= 0 by A2, SERIES_3:34;
then  abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= (((Partial_Sums (abs seq)) . n) - ((Partial_Sums (abs seq)) . m)) + ((Partial_Sums (abs seq)) . m) by A3, A1, SERIES_1:34, XREAL_1:38;
hence  ( abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= ((Partial_Sums (abs seq)) . n) - ((Partial_Sums (abs seq)) . m) & abs (((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)) <= (Partial_Sums (abs seq)) . n ) by A3, A1, SERIES_1:34; ::_thesis: verum
end;

theorem Th11: :: LPSPACE2:11
for seq, seq2 being Real_Sequence
for k being positive Real st seq is convergent & ( for n being Element of NAT holds seq2 . n = |.((lim seq) - (seq . n)).| to_power k ) holds
( seq2 is convergent & lim seq2 = 0 )
proof
let seq, seq2 be Real_Sequence; ::_thesis: for k being positive Real st seq is convergent & ( for n being Element of NAT holds seq2 . n = |.((lim seq) - (seq . n)).| to_power k ) holds
( seq2 is convergent & lim seq2 = 0 )
let k be positive Real; ::_thesis: ( seq is convergent & ( for n being Element of NAT holds seq2 . n = |.((lim seq) - (seq . n)).| to_power k ) implies ( seq2 is convergent & lim seq2 = 0 ) )
set r = lim seq;
assume A1: ( seq is convergent & ( for n being Element of NAT holds seq2 . n = |.((lim seq) - (seq . n)).| to_power k ) ) ; ::_thesis: ( seq2 is convergent & lim seq2 = 0 )
deffunc H1( Element of NAT ) ->  Element of REAL = |.((lim seq) - (seq . $1)).|;
consider seq1 being Real_Sequence such that
A2: for n being Element of NAT holds seq1 . n = H1(n) from SEQ_1:sch_1();
deffunc H2( Element of NAT ) ->  Element of REAL = lim seq;
consider seq0 being Real_Sequence such that
A3: for n being Element of NAT holds seq0 . n = H2(n) from SEQ_1:sch_1();
 for n being Nat holds seq0 . n = lim seq
proof
let n be Nat; ::_thesis: seq0 . n = lim seq
 n in NAT by ORDINAL1:def_12;
hence  seq0 . n = lim seq by A3; ::_thesis: verum
end;
then A4: seq0 is constant by VALUED_0:def_18;
then A5: seq0 - seq is convergent by A1;
A6: ( dom seq0 = NAT & dom seq = NAT & dom (seq0 - seq) = NAT & dom seq1 = NAT ) by FUNCT_2:def_1;
A7: dom (abs (seq0 - seq)) = dom (seq0 - seq) by VALUED_1:def_11;
 for n being Element of NAT holds (abs (seq0 - seq)) . n = seq1 . n
proof
let n be Element of NAT ; ::_thesis: (abs (seq0 - seq)) . n = seq1 . n
 seq1 . n = |.((lim seq) - (seq . n)).| by A2;
then  seq1 . n = |.((seq0 . n) - (seq . n)).| by A3;
then  seq1 . n = abs ((seq0 - seq) . n) by A6, VALUED_1:13;
hence  (abs (seq0 - seq)) . n = seq1 . n by A6, A7, VALUED_1:def_11; ::_thesis: verum
end;
then A8: abs (seq0 - seq) = seq1 by FUNCT_2:63;
then A9: seq1 is convergent by A5;
 lim (seq0 - seq) = (seq0 . 0) - (lim seq) by A4, A1, SEQ_4:42;
then  lim (seq0 - seq) = (lim seq) - (lim seq) by A3;
then A10: lim seq1 = 0 by A5, A8, COMPLEX1:44, SEQ_4:14;
 for n being Element of NAT holds
( seq2 . n = (seq1 . n) to_power k & seq1 . n >= 0 )
proof
let n be Element of NAT ; ::_thesis: ( seq2 . n = (seq1 . n) to_power k & seq1 . n >= 0 )
 |.((lim seq) - (seq . n)).| = seq1 . n by A2;
hence  ( seq2 . n = (seq1 . n) to_power k & seq1 . n >= 0 ) by A1, COMPLEX1:46; ::_thesis: verum
end;
then  ( seq2 is convergent & lim seq2 = (lim seq1) to_power k ) by A9, HOLDER_1:10;
hence  ( seq2 is convergent & lim seq2 = 0 ) by A10, POWER:def_2; ::_thesis: verum
end;

Lm1:  for seq being Real_Sequence
for n being Element of NAT holds abs ((Partial_Sums seq) . n) <= (Partial_Sums (abs seq)) . n
by NAGATA_2:13;

begin

theorem Th12: :: LPSPACE2:12
for k being positive Real
for X being non empty set holds (X --> 0) to_power k = X --> 0
proof
let k be positive Real; ::_thesis: for X being non empty set holds (X --> 0) to_power k = X --> 0
let X be non empty set ; ::_thesis: (X --> 0) to_power k = X --> 0
A1: dom ((X --> 0) to_power k) = dom (X --> 0) by MESFUN6C:def_4;
now__::_thesis:_for_x_being_Element_of_X_st_x_in_dom_((X_-->_0)_to_power_k)_holds_
((X_-->_0)_to_power_k)_._x_=_(X_-->_0)_._x
let x be Element of X; ::_thesis: ( x in dom ((X --> 0) to_power k) implies ((X --> 0) to_power k) . x = (X --> 0) . x )
assume  x in dom ((X --> 0) to_power k) ; ::_thesis: ((X --> 0) to_power k) . x = (X --> 0) . x
then  ((X --> 0) to_power k) . x = ((X --> 0) . x) to_power k by MESFUN6C:def_4;
then  ((X --> 0) to_power k) . x = 0 to_power k by FUNCOP_1:7;
then  ((X --> 0) to_power k) . x = 0 by POWER:def_2;
hence  ((X --> 0) to_power k) . x = (X --> 0) . x by FUNCOP_1:7; ::_thesis: verum
end;
hence  (X --> 0) to_power k = X --> 0 by A1, PARTFUN1:5; ::_thesis: verum
end;

theorem Th13: :: LPSPACE2:13
for X being non empty set
for f being PartFunc of X,REAL
for D being set holds abs (f | D) = (abs f) | D
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL
for D being set holds abs (f | D) = (abs f) | D
let f be PartFunc of X,REAL; ::_thesis: for D being set holds abs (f | D) = (abs f) | D
let D be set ; ::_thesis: abs (f | D) = (abs f) | D
A1: dom (abs (f | D)) = dom (f | D) by VALUED_1:def_11;
then  dom (abs (f | D)) = (dom f) /\ D by RELAT_1:61;
then  dom (abs (f | D)) = (dom (abs f)) /\ D by VALUED_1:def_11;
then A2: dom (abs (f | D)) = dom ((abs f) | D) by RELAT_1:61;
 for x being Element of X st x in dom (abs (f | D)) holds
(abs (f | D)) . x = ((abs f) | D) . x
proof
let x be Element of X; ::_thesis: ( x in dom (abs (f | D)) implies (abs (f | D)) . x = ((abs f) | D) . x )
assume A3: x in dom (abs (f | D)) ; ::_thesis: (abs (f | D)) . x = ((abs f) | D) . x
then  x in dom f by A1, RELAT_1:57;
then A4: x in dom (abs f) by VALUED_1:def_11;
 (abs (f | D)) . x = abs ((f | D) . x) by A3, VALUED_1:def_11;
then  (abs (f | D)) . x = abs (f . x) by A3, A1, FUNCT_1:47;
then  (abs (f | D)) . x = (abs f) . x by A4, VALUED_1:def_11;
hence  (abs (f | D)) . x = ((abs f) | D) . x by A3, A2, FUNCT_1:47; ::_thesis: verum
end;
hence  abs (f | D) = (abs f) | D by A2, PARTFUN1:5; ::_thesis: verum
end;

registration
let X be non empty set ;
let f be PartFunc of X,REAL;
cluster|.f.| ->  nonnegative ;
coherence
 abs f is nonnegative
proof
now__::_thesis:_for_x_being_set_st_x_in_dom_(abs_f)_holds_
0_<=_(abs_f)_._x
let x be set ; ::_thesis: ( x in dom (abs f) implies 0 <= (abs f) . x )
assume  x in dom (abs f) ; ::_thesis: 0 <= (abs f) . x
then  (abs f) . x = abs (f . x) by VALUED_1:def_11;
hence  0 <= (abs f) . x by COMPLEX1:46; ::_thesis: verum
end;
hence  abs f is nonnegative by MESFUNC6:52; ::_thesis: verum
end;
end;

theorem Th14: :: LPSPACE2:14
for X being non empty set
for f being PartFunc of X,REAL st f is nonnegative holds
abs f = f
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL st f is nonnegative holds
abs f = f
let f be PartFunc of X,REAL; ::_thesis: ( f is nonnegative implies abs f = f )
A1: dom f = dom (abs f) by VALUED_1:def_11;
assume A2: f is nonnegative ; ::_thesis: abs f = f
now__::_thesis:_for_x_being_Element_of_X_st_x_in_dom_f_holds_
(abs_f)_._x_=_f_._x
let x be Element of X; ::_thesis: ( x in dom f implies (abs f) . x = f . x )
A3: f . x >= 0 by A2, MESFUNC6:51;
assume  x in dom f ; ::_thesis: (abs f) . x = f . x
then  x in dom (abs f) by VALUED_1:def_11;
then  (abs f) . x = abs (f . x) by VALUED_1:def_11;
hence  (abs f) . x = f . x by A3, ABSVALUE:def_1; ::_thesis: verum
end;
hence  abs f = f by A1, PARTFUN1:5; ::_thesis: verum
end;

theorem Th15: :: LPSPACE2:15
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds
0 = f . x ) holds
( f is_integrable_on M & Integral (M,f) = 0 )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds
0 = f . x ) holds
( f is_integrable_on M & Integral (M,f) = 0 )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds
0 = f . x ) holds
( f is_integrable_on M & Integral (M,f) = 0 )
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds
0 = f . x ) holds
( f is_integrable_on M & Integral (M,f) = 0 )
let f be PartFunc of X,REAL; ::_thesis: ( X = dom f & ( for x being Element of X st x in dom f holds
0 = f . x ) implies ( f is_integrable_on M & Integral (M,f) = 0 ) )
assume A1: ( X = dom f & ( for x being Element of X st x in dom f holds
0 = f . x ) ) ; ::_thesis: ( f is_integrable_on M & Integral (M,f) = 0 )
 X is Element of S by MEASURE1:7;
then  ( R_EAL f is_integrable_on M & Integral (M,(R_EAL f)) = 0 ) by A1, LPSPACE1:22;
hence  ( f is_integrable_on M & Integral (M,f) = 0 ) by MESFUNC6:def_4; ::_thesis: verum
end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func Lp_Functions (M,k) ->  non empty Subset of (RLSp_PFunct X) equals :: LPSPACE2:def 2
 {  f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )  }  ;
correctness
coherence
{  f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )  }  is non empty Subset of (RLSp_PFunct X);
proof
set V =  {  f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )  }  ;
A1:  {  f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )  }  c= PFuncs (X,REAL)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )  }  or x in PFuncs (X,REAL) )
assume  x in  {  f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )  }  ; ::_thesis: x in PFuncs (X,REAL)
then  ex f being PartFunc of X,REAL st
( x = f & ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M ) ) ;
hence  x in PFuncs (X,REAL) by PARTFUN1:45; ::_thesis: verum
end;
reconsider g = X --> 0 as Function of X,REAL by FUNCOP_1:46;
reconsider Ef = X as Element of S by MEASURE1:34;
set h = (abs g) to_power k;
A2: dom g = X by FUNCOP_1:13;
 for x being set st x in dom g holds
g . x = 0 by FUNCOP_1:7;
then A3: g is_measurable_on Ef by A2, LPSPACE1:52;
 Ef ` = {} by XBOOLE_1:37;
then A4: M . (Ef `) = 0 by VALUED_0:def_19;
 for x being set st x in dom (X --> 0) holds
0 <= (X --> 0) . x ;
then  abs g = X --> 0 by Th14, MESFUNC6:52;
then A5: (abs g) to_power k = g by Th12;
then  for x being Element of X st x in dom ((abs g) to_power k) holds
((abs g) to_power k) . x = 0 by FUNCOP_1:7;
then  (abs g) to_power k is_integrable_on M by Th15, A5, A2;
then  g in  {  f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )  }  by A3, A4, A2;
hence   {  f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )  }  is non empty Subset of (RLSp_PFunct X) by A1; ::_thesis: verum
end;
end;

:: deftheorem  defines Lp_Functions LPSPACE2:def_2_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds Lp_Functions (M,k) =  {  f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )  }  ;

theorem Th16: :: LPSPACE2:16
for a, b, k being Real st k > 0 holds
( (abs (a + b)) to_power k <= ((abs a) + (abs b)) to_power k & ((abs a) + (abs b)) to_power k <= (2 * (max ((abs a),(abs b)))) to_power k & (abs (a + b)) to_power k <= (2 * (max ((abs a),(abs b)))) to_power k )
proof
let a, b, k be Real; ::_thesis: ( k > 0 implies ( (abs (a + b)) to_power k <= ((abs a) + (abs b)) to_power k & ((abs a) + (abs b)) to_power k <= (2 * (max ((abs a),(abs b)))) to_power k & (abs (a + b)) to_power k <= (2 * (max ((abs a),(abs b)))) to_power k ) )
assume A1: k > 0 ; ::_thesis: ( (abs (a + b)) to_power k <= ((abs a) + (abs b)) to_power k & ((abs a) + (abs b)) to_power k <= (2 * (max ((abs a),(abs b)))) to_power k & (abs (a + b)) to_power k <= (2 * (max ((abs a),(abs b)))) to_power k )
A2: abs (a + b) <= (abs a) + (abs b) by ABSVALUE:9;
 ( abs a <= max ((abs a),(abs b)) & abs b <= max ((abs a),(abs b)) ) by XXREAL_0:25;
then A3: (abs a) + (abs b) <= (max ((abs a),(abs b))) + (max ((abs a),(abs b))) by XREAL_1:7;
then A4: abs (a + b) <= 2 * (max ((abs a),(abs b))) by A2, XXREAL_0:2;
 0 <= abs (a + b) by COMPLEX1:46;
hence  ( (abs (a + b)) to_power k <= ((abs a) + (abs b)) to_power k & ((abs a) + (abs b)) to_power k <= (2 * (max ((abs a),(abs b)))) to_power k & (abs (a + b)) to_power k <= (2 * (max ((abs a),(abs b)))) to_power k ) by A1, A2, A3, A4, HOLDER_1:3; ::_thesis: verum
end;

theorem Th17: :: LPSPACE2:17
for a, b, k being Real st a >= 0 & b >= 0 & k > 0 holds
(max (a,b)) to_power k <= (a to_power k) + (b to_power k)
proof
let a, b, k be Real; ::_thesis: ( a >= 0 & b >= 0 & k > 0 implies (max (a,b)) to_power k <= (a to_power k) + (b to_power k) )
assume A1: ( a >= 0 & b >= 0 & k > 0 ) ; ::_thesis: (max (a,b)) to_power k <= (a to_power k) + (b to_power k)
percases ( ( a <> 0 & b <> 0 ) or a = 0 or b = 0 ) ;
suppose ( a <> 0 & b <> 0 ) ; ::_thesis: (max (a,b)) to_power k <= (a to_power k) + (b to_power k)
then A2: ( a to_power k >= 0 & b to_power k >= 0 ) by A1, POWER:34;
 ( max (a,b) = a or max (a,b) = b ) by XXREAL_0:def_10;
hence  (max (a,b)) to_power k <= (a to_power k) + (b to_power k) by A2, XREAL_1:40; ::_thesis: verum
end;
supposeA3: a = 0 ; ::_thesis: (max (a,b)) to_power k <= (a to_power k) + (b to_power k)
then  a to_power k = 0 by A1, POWER:def_2;
hence  (max (a,b)) to_power k <= (a to_power k) + (b to_power k) by A1, A3, XXREAL_0:def_10; ::_thesis: verum
end;
supposeA4: b = 0 ; ::_thesis: (max (a,b)) to_power k <= (a to_power k) + (b to_power k)
then  b to_power k = 0 by A1, POWER:def_2;
hence  (max (a,b)) to_power k <= (a to_power k) + (b to_power k) by A1, A4, XXREAL_0:def_10; ::_thesis: verum
end;
end;
end;

theorem Th18: :: LPSPACE2:18
for X being non empty set
for f being PartFunc of X,REAL
for a, b being Real st b > 0 holds
((abs a) to_power b) (#) ((abs f) to_power b) = (abs (a (#) f)) to_power b
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL
for a, b being Real st b > 0 holds
((abs a) to_power b) (#) ((abs f) to_power b) = (abs (a (#) f)) to_power b
let f be PartFunc of X,REAL; ::_thesis: for a, b being Real st b > 0 holds
((abs a) to_power b) (#) ((abs f) to_power b) = (abs (a (#) f)) to_power b
let a, b be Real; ::_thesis: ( b > 0 implies ((abs a) to_power b) (#) ((abs f) to_power b) = (abs (a (#) f)) to_power b )
assume A1: b > 0 ; ::_thesis: ((abs a) to_power b) (#) ((abs f) to_power b) = (abs (a (#) f)) to_power b
A2: ( dom (((abs a) to_power b) (#) ((abs f) to_power b)) = dom ((abs f) to_power b) & dom (a (#) f) = dom f ) by VALUED_1:def_5;
A3: ( dom ((abs f) to_power b) = dom (abs f) & dom (abs (a (#) f)) = dom ((abs (a (#) f)) to_power b) ) by MESFUN6C:def_4;
A4: ( dom (abs f) = dom f & dom (abs (a (#) f)) = dom (a (#) f) ) by VALUED_1:def_11;
 for x being Element of X st x in dom (((abs a) to_power b) (#) ((abs f) to_power b)) holds
(((abs a) to_power b) (#) ((abs f) to_power b)) . x = ((abs (a (#) f)) to_power b) . x
proof
let x be Element of X; ::_thesis: ( x in dom (((abs a) to_power b) (#) ((abs f) to_power b)) implies (((abs a) to_power b) (#) ((abs f) to_power b)) . x = ((abs (a (#) f)) to_power b) . x )
assume A5: x in dom (((abs a) to_power b) (#) ((abs f) to_power b)) ; ::_thesis: (((abs a) to_power b) (#) ((abs f) to_power b)) . x = ((abs (a (#) f)) to_power b) . x
A6: ( abs (f . x) >= 0 & abs a >= 0 ) by COMPLEX1:46;
(((abs a) to_power b) (#) ((abs f) to_power b)) . x = ((abs a) to_power b) * (((abs f) to_power b) . x) by A5, VALUED_1:def_5
.= ((abs a) to_power b) * (((abs f) . x) to_power b) by A2, A5, MESFUN6C:def_4
.= ((abs a) to_power b) * ((abs (f . x)) to_power b) by VALUED_1:18
.= ((abs a) * (abs (f . x))) to_power b by A1, A6, Th5
.= (abs (a * (f . x))) to_power b by COMPLEX1:65
.= (abs ((a (#) f) . x)) to_power b by VALUED_1:6
.= ((abs (a (#) f)) . x) to_power b by VALUED_1:18 ;
hence  (((abs a) to_power b) (#) ((abs f) to_power b)) . x = ((abs (a (#) f)) to_power b) . x by A2, A3, A4, A5, MESFUN6C:def_4; ::_thesis: verum
end;
hence  ((abs a) to_power b) (#) ((abs f) to_power b) = (abs (a (#) f)) to_power b by A2, A3, A4, PARTFUN1:5; ::_thesis: verum
end;

theorem Th19: :: LPSPACE2:19
for X being non empty set
for f being PartFunc of X,REAL
for a, b being Real st a > 0 & b > 0 holds
(a to_power b) (#) ((abs f) to_power b) = (a (#) (abs f)) to_power b
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL
for a, b being Real st a > 0 & b > 0 holds
(a to_power b) (#) ((abs f) to_power b) = (a (#) (abs f)) to_power b
let f be PartFunc of X,REAL; ::_thesis: for a, b being Real st a > 0 & b > 0 holds
(a to_power b) (#) ((abs f) to_power b) = (a (#) (abs f)) to_power b
let a, b be Real; ::_thesis: ( a > 0 & b > 0 implies (a to_power b) (#) ((abs f) to_power b) = (a (#) (abs f)) to_power b )
assume A1: ( a > 0 & b > 0 ) ; ::_thesis: (a to_power b) (#) ((abs f) to_power b) = (a (#) (abs f)) to_power b
then A2: abs a = a by COMPLEX1:43;
then  (a to_power b) (#) ((abs f) to_power b) = (abs (a (#) f)) to_power b by A1, Th18;
hence  (a to_power b) (#) ((abs f) to_power b) = (a (#) (abs f)) to_power b by A2, RFUNCT_1:25; ::_thesis: verum
end;

theorem Th20: :: LPSPACE2:20
for X being non empty set
for f being PartFunc of X,REAL
for k being real number
for E being set holds (f | E) to_power k = (f to_power k) | E
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL
for k being real number
for E being set holds (f | E) to_power k = (f to_power k) | E
let f be PartFunc of X,REAL; ::_thesis: for k being real number
for E being set holds (f | E) to_power k = (f to_power k) | E
let k be real number ; ::_thesis: for E being set holds (f | E) to_power k = (f to_power k) | E
let E be set ; ::_thesis: (f | E) to_power k = (f to_power k) | E
A1: dom ((f | E) to_power k) = dom (f | E) by MESFUN6C:def_4;
then  dom ((f | E) to_power k) = (dom f) /\ E by RELAT_1:61;
then A2: dom ((f | E) to_power k) = (dom (f to_power k)) /\ E by MESFUN6C:def_4;
then A3: dom ((f | E) to_power k) = dom ((f to_power k) | E) by RELAT_1:61;
now__::_thesis:_for_x_being_Element_of_X_st_x_in_dom_((f_|_E)_to_power_k)_holds_
((f_|_E)_to_power_k)_._x_=_((f_to_power_k)_|_E)_._x
let x be Element of X; ::_thesis: ( x in dom ((f | E) to_power k) implies ((f | E) to_power k) . x = ((f to_power k) | E) . x )
assume A4: x in dom ((f | E) to_power k) ; ::_thesis: ((f | E) to_power k) . x = ((f to_power k) | E) . x
then  ((f | E) to_power k) . x = ((f | E) . x) to_power k by MESFUN6C:def_4;
then A5: ((f | E) to_power k) . x = (f . x) to_power k by A1, A4, FUNCT_1:47;
 x in dom (f to_power k) by A2, A4, XBOOLE_0:def_4;
then  ((f | E) to_power k) . x = (f to_power k) . x by A5, MESFUN6C:def_4;
hence  ((f | E) to_power k) . x = ((f to_power k) | E) . x by A4, A3, FUNCT_1:47; ::_thesis: verum
end;
hence  (f | E) to_power k = (f to_power k) | E by A3, PARTFUN1:5; ::_thesis: verum
end;

theorem Th21: :: LPSPACE2:21
for a, b, k being Real st k > 0 holds
(abs (a + b)) to_power k <= (2 to_power k) * (((abs a) to_power k) + ((abs b) to_power k))
proof
let a, b, k be Real; ::_thesis: ( k > 0 implies (abs (a + b)) to_power k <= (2 to_power k) * (((abs a) to_power k) + ((abs b) to_power k)) )
assume A1: k > 0 ; ::_thesis: (abs (a + b)) to_power k <= (2 to_power k) * (((abs a) to_power k) + ((abs b) to_power k))
then A2: (abs (a + b)) to_power k <= (2 * (max ((abs a),(abs b)))) to_power k by Th16;
A3: ( abs a >= 0 & abs b >= 0 ) by COMPLEX1:46;
then A4: (max ((abs a),(abs b))) to_power k <= ((abs a) to_power k) + ((abs b) to_power k) by A1, Th17;
 ( max ((abs a),(abs b)) = abs a or max ((abs a),(abs b)) = abs b ) by XXREAL_0:16;
then A5: (2 * (max ((abs a),(abs b)))) to_power k = (2 to_power k) * ((max ((abs a),(abs b))) to_power k) by A1, A3, Th5;
 2 to_power k > 0 by POWER:34;
then  (2 to_power k) * ((max ((abs a),(abs b))) to_power k) <= (2 to_power k) * (((abs a) to_power k) + ((abs b) to_power k)) by A4, XREAL_1:64;
hence  (abs (a + b)) to_power k <= (2 to_power k) * (((abs a) to_power k) + ((abs b) to_power k)) by A2, A5, XXREAL_0:2; ::_thesis: verum
end;

theorem Th22: :: LPSPACE2:22
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for f, g being PartFunc of X,REAL st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
( (abs f) to_power k is_integrable_on M & (abs g) to_power k is_integrable_on M & ((abs f) to_power k) + ((abs g) to_power k) is_integrable_on M )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for f, g being PartFunc of X,REAL st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
( (abs f) to_power k is_integrable_on M & (abs g) to_power k is_integrable_on M & ((abs f) to_power k) + ((abs g) to_power k) is_integrable_on M )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real
for f, g being PartFunc of X,REAL st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
( (abs f) to_power k is_integrable_on M & (abs g) to_power k is_integrable_on M & ((abs f) to_power k) + ((abs g) to_power k) is_integrable_on M )
let M be sigma_Measure of S; ::_thesis: for k being positive Real
for f, g being PartFunc of X,REAL st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
( (abs f) to_power k is_integrable_on M & (abs g) to_power k is_integrable_on M & ((abs f) to_power k) + ((abs g) to_power k) is_integrable_on M )
let k be positive Real; ::_thesis: for f, g being PartFunc of X,REAL st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
( (abs f) to_power k is_integrable_on M & (abs g) to_power k is_integrable_on M & ((abs f) to_power k) + ((abs g) to_power k) is_integrable_on M )
let f, g be PartFunc of X,REAL; ::_thesis: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) implies ( (abs f) to_power k is_integrable_on M & (abs g) to_power k is_integrable_on M & ((abs f) to_power k) + ((abs g) to_power k) is_integrable_on M ) )
assume A1: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) ; ::_thesis: ( (abs f) to_power k is_integrable_on M & (abs g) to_power k is_integrable_on M & ((abs f) to_power k) + ((abs g) to_power k) is_integrable_on M )
then A2: ex f1 being PartFunc of X,REAL st
( f = f1 & ex Ev being Element of S st
( M . (Ev `) = 0 & dom f1 = Ev & f1 is_measurable_on Ev & (abs f1) to_power k is_integrable_on M ) ) ;
 ex g1 being PartFunc of X,REAL st
( g = g1 & ex Eu being Element of S st
( M . (Eu `) = 0 & dom g1 = Eu & g1 is_measurable_on Eu & (abs g1) to_power k is_integrable_on M ) ) by A1;
hence  ( (abs f) to_power k is_integrable_on M & (abs g) to_power k is_integrable_on M & ((abs f) to_power k) + ((abs g) to_power k) is_integrable_on M ) by A2, MESFUNC6:100; ::_thesis: verum
end;

theorem Th23: :: LPSPACE2:23
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( X --> 0 is PartFunc of X,REAL & X --> 0 in Lp_Functions (M,k) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( X --> 0 is PartFunc of X,REAL & X --> 0 in Lp_Functions (M,k) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real holds
( X --> 0 is PartFunc of X,REAL & X --> 0 in Lp_Functions (M,k) )
let M be sigma_Measure of S; ::_thesis: for k being positive Real holds
( X --> 0 is PartFunc of X,REAL & X --> 0 in Lp_Functions (M,k) )
let k be positive Real; ::_thesis: ( X --> 0 is PartFunc of X,REAL & X --> 0 in Lp_Functions (M,k) )
reconsider g = X --> 0 as Function of X,REAL by FUNCOP_1:46;
reconsider ND = X as Element of S by MEASURE1:34;
 ND ` = {} by XBOOLE_1:37;
then A1: M . (ND `) = 0 by VALUED_0:def_19;
A2: dom g = X by FUNCT_2:def_1;
 for x being Element of X st x in dom g holds
g . x = 0 by FUNCOP_1:7;
then A3: g is_integrable_on M by A2, Th15;
 for x being set st x in dom g holds
0 <= g . x ;
then  abs g = g by Th14, MESFUNC6:52;
then A4: (abs g) to_power k = g by Th12;
 for x being set st x in dom g holds
g . x = 0 by FUNCOP_1:7;
then  g is_measurable_on ND by A2, LPSPACE1:52;
hence  ( X --> 0 is PartFunc of X,REAL & X --> 0 in Lp_Functions (M,k) ) by A1, A2, A3, A4; ::_thesis: verum
end;

theorem Th24: :: LPSPACE2:24
for X being non empty set
for k being Real st k > 0 holds
for f, g being PartFunc of X,REAL
for x being Element of X st x in (dom f) /\ (dom g) holds
((abs (f + g)) to_power k) . x <= ((2 to_power k) (#) (((abs f) to_power k) + ((abs g) to_power k))) . x
proof
let X be non empty set ; ::_thesis: for k being Real st k > 0 holds
for f, g being PartFunc of X,REAL
for x being Element of X st x in (dom f) /\ (dom g) holds
((abs (f + g)) to_power k) . x <= ((2 to_power k) (#) (((abs f) to_power k) + ((abs g) to_power k))) . x
let k be Element of REAL ; ::_thesis: ( k > 0 implies for f, g being PartFunc of X,REAL
for x being Element of X st x in (dom f) /\ (dom g) holds
((abs (f + g)) to_power k) . x <= ((2 to_power k) (#) (((abs f) to_power k) + ((abs g) to_power k))) . x )
assume A1: k > 0 ; ::_thesis: for f, g being PartFunc of X,REAL
for x being Element of X st x in (dom f) /\ (dom g) holds
((abs (f + g)) to_power k) . x <= ((2 to_power k) (#) (((abs f) to_power k) + ((abs g) to_power k))) . x
let f, g be PartFunc of X,REAL; ::_thesis: for x being Element of X st x in (dom f) /\ (dom g) holds
((abs (f + g)) to_power k) . x <= ((2 to_power k) (#) (((abs f) to_power k) + ((abs g) to_power k))) . x
let x be Element of X; ::_thesis: ( x in (dom f) /\ (dom g) implies ((abs (f + g)) to_power k) . x <= ((2 to_power k) (#) (((abs f) to_power k) + ((abs g) to_power k))) . x )
assume A2: x in (dom f) /\ (dom g) ; ::_thesis: ((abs (f + g)) to_power k) . x <= ((2 to_power k) (#) (((abs f) to_power k) + ((abs g) to_power k))) . x
A3: dom (f + g) = (dom f) /\ (dom g) by VALUED_1:def_1;
then  dom (abs (f + g)) = (dom f) /\ (dom g) by VALUED_1:def_11;
then  x in dom ((abs (f + g)) to_power k) by A2, MESFUN6C:def_4;
then A4: ((abs (f + g)) to_power k) . x = ((abs (f + g)) . x) to_power k by MESFUN6C:def_4
.= (abs ((f + g) . x)) to_power k by VALUED_1:18
.= (abs ((f . x) + (g . x))) to_power k by A3, A2, VALUED_1:def_1 ;
 ( dom (abs f) = dom f & dom (abs g) = dom g ) by VALUED_1:def_11;
then  ( x in dom (abs f) & x in dom (abs g) ) by A2, XBOOLE_0:def_4;
then A5: ( x in dom ((abs f) to_power k) & x in dom ((abs g) to_power k) ) by MESFUN6C:def_4;
 ( (abs (f . x)) to_power k = ((abs f) . x) to_power k & (abs (g . x)) to_power k = ((abs g) . x) to_power k ) by VALUED_1:18;
then A6: ( (abs (f . x)) to_power k = ((abs f) to_power k) . x & (abs (g . x)) to_power k = ((abs g) to_power k) . x ) by A5, MESFUN6C:def_4;
 dom (((abs f) to_power k) + ((abs g) to_power k)) = (dom ((abs f) to_power k)) /\ (dom ((abs g) to_power k)) by VALUED_1:def_1;
then  x in dom (((abs f) to_power k) + ((abs g) to_power k)) by A5, XBOOLE_0:def_4;
then (2 to_power k) * (((abs (f . x)) to_power k) + ((abs (g . x)) to_power k)) = (2 to_power k) * ((((abs f) to_power k) + ((abs g) to_power k)) . x) by A6, VALUED_1:def_1
.= ((2 to_power k) (#) (((abs f) to_power k) + ((abs g) to_power k))) . x by VALUED_1:6 ;
hence  ((abs (f + g)) to_power k) . x <= ((2 to_power k) (#) (((abs f) to_power k) + ((abs g) to_power k))) . x by A1, A4, Th21; ::_thesis: verum
end;

theorem Th25: :: LPSPACE2:25
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f + g in Lp_Functions (M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f + g in Lp_Functions (M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f + g in Lp_Functions (M,k)
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f + g in Lp_Functions (M,k)
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f + g in Lp_Functions (M,k)
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) implies f + g in Lp_Functions (M,k) )
set W = Lp_Functions (M,k);
assume A1: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) ; ::_thesis: f + g in Lp_Functions (M,k)
then consider f1 being PartFunc of X,REAL such that
A2: ( f1 = f & ex Ef1 being Element of S st
( M . (Ef1 `) = 0 & dom f1 = Ef1 & f1 is_measurable_on Ef1 & (abs f1) to_power k is_integrable_on M ) ) ;
consider Ef being Element of S such that
A3: ( M . (Ef `) = 0 & dom f1 = Ef & f1 is_measurable_on Ef & (abs f1) to_power k is_integrable_on M ) by A2;
consider g1 being PartFunc of X,REAL such that
A4: ( g1 = g & ex Eg1 being Element of S st
( M . (Eg1 `) = 0 & dom g1 = Eg1 & g1 is_measurable_on Eg1 & (abs g1) to_power k is_integrable_on M ) ) by A1;
consider Eg being Element of S such that
A5: ( M . (Eg `) = 0 & dom g1 = Eg & g1 is_measurable_on Eg & (abs g1) to_power k is_integrable_on M ) by A4;
A6: dom (f1 + g1) = Ef /\ Eg by A3, A5, VALUED_1:def_1;
set Efg = Ef /\ Eg;
set s = (abs (f1 + g1)) to_power k;
set t = (2 to_power k) (#) (((abs f1) to_power k) + ((abs g1) to_power k));
A7: (Ef /\ Eg) ` = (X \ Ef) \/ (X \ Eg) by XBOOLE_1:54;
 ( Ef ` is Element of S & Eg ` is Element of S ) by MEASURE1:34;
then  ( Ef ` is measure_zero of M & Eg ` is measure_zero of M ) by A3, A5, MEASURE1:def_7;
then  (Ef `) \/ (Eg `) is measure_zero of M by MEASURE1:37;
then A8: M . ((Ef /\ Eg) `) = 0 by A7, MEASURE1:def_7;
 ( f1 is_measurable_on Ef /\ Eg & g1 is_measurable_on Ef /\ Eg ) by A3, A5, MESFUNC6:16, XBOOLE_1:17;
then A9: f1 + g1 is_measurable_on Ef /\ Eg by MESFUNC6:26;
then A10: abs (f1 + g1) is_measurable_on Ef /\ Eg by A6, MESFUNC6:48;
 ((abs f1) to_power k) + ((abs g1) to_power k) is_integrable_on M by A1, A2, A4, Th22;
then A11: (2 to_power k) (#) (((abs f1) to_power k) + ((abs g1) to_power k)) is_integrable_on M by MESFUNC6:102;
A12: ( dom (abs f1) = dom f1 & dom (abs g1) = dom g1 & dom (abs (f1 + g1)) = dom (f1 + g1) ) by VALUED_1:def_11;
then A13: (abs (f1 + g1)) to_power k is_measurable_on Ef /\ Eg by A6, A10, MESFUN6C:29;
A14: abs ((abs (f1 + g1)) to_power k) = (abs (f1 + g1)) to_power k by Th14;
A15: dom ((abs (f1 + g1)) to_power k) = Ef /\ Eg by A6, A12, MESFUN6C:def_4;
A16: dom ((2 to_power k) (#) (((abs f1) to_power k) + ((abs g1) to_power k))) = dom (((abs f1) to_power k) + ((abs g1) to_power k)) by VALUED_1:def_5
.= (dom ((abs f1) to_power k)) /\ (dom ((abs g1) to_power k)) by VALUED_1:def_1
.= (dom (abs f1)) /\ (dom ((abs g1) to_power k)) by MESFUN6C:def_4
.= (dom (abs f1)) /\ (dom (abs g1)) by MESFUN6C:def_4
.= dom (f1 + g1) by A12, VALUED_1:def_1
.= dom ((abs (f1 + g1)) to_power k) by A12, MESFUN6C:def_4 ;
now__::_thesis:_for_x_being_Element_of_X_st_x_in_dom_((abs_(f1_+_g1))_to_power_k)_holds_
abs_(((abs_(f1_+_g1))_to_power_k)_._x)_<=_((2_to_power_k)_(#)_(((abs_f1)_to_power_k)_+_((abs_g1)_to_power_k)))_._x
let x be Element of X; ::_thesis: ( x in dom ((abs (f1 + g1)) to_power k) implies abs (((abs (f1 + g1)) to_power k) . x) <= ((2 to_power k) (#) (((abs f1) to_power k) + ((abs g1) to_power k))) . x )
assume  x in dom ((abs (f1 + g1)) to_power k) ; ::_thesis: abs (((abs (f1 + g1)) to_power k) . x) <= ((2 to_power k) (#) (((abs f1) to_power k) + ((abs g1) to_power k))) . x
then  (abs ((abs (f1 + g1)) to_power k)) . x <= ((2 to_power k) (#) (((abs f1) to_power k) + ((abs g1) to_power k))) . x by A14, Th24, A3, A5, A15;
hence  abs (((abs (f1 + g1)) to_power k) . x) <= ((2 to_power k) (#) (((abs f1) to_power k) + ((abs g1) to_power k))) . x by VALUED_1:18; ::_thesis: verum
end;
then  (abs (f1 + g1)) to_power k is_integrable_on M by A13, A15, A16, A11, MESFUNC6:96;
hence  f + g in Lp_Functions (M,k) by A2, A4, A8, A6, A9; ::_thesis: verum
end;

theorem Th26: :: LPSPACE2:26
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let f be PartFunc of X,REAL; ::_thesis: for a being Real
for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let a be Real; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) implies a (#) f in Lp_Functions (M,k) )
assume  f in Lp_Functions (M,k) ; ::_thesis: a (#) f in Lp_Functions (M,k)
then consider f1 being PartFunc of X,REAL such that
A1: ( f1 = f & ex Ef1 being Element of S st
( M . (Ef1 `) = 0 & dom f1 = Ef1 & f1 is_measurable_on Ef1 & (abs f1) to_power k is_integrable_on M ) ) ;
consider Ef being Element of S such that
A2: ( M . (Ef `) = 0 & dom f1 = Ef & f1 is_measurable_on Ef & (abs f1) to_power k is_integrable_on M ) by A1;
A3: ( dom (a (#) f1) = Ef & a (#) f1 is_measurable_on Ef ) by A2, MESFUNC6:21, VALUED_1:def_5;
 ((abs a) to_power k) (#) ((abs f1) to_power k) is_integrable_on M by A1, MESFUNC6:102;
then  (abs (a (#) f1)) to_power k is_integrable_on M by Th18;
hence  a (#) f in Lp_Functions (M,k) by A1, A2, A3; ::_thesis: verum
end;

theorem Th27: :: LPSPACE2:27
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f - g in Lp_Functions (M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f - g in Lp_Functions (M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f - g in Lp_Functions (M,k)
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f - g in Lp_Functions (M,k)
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds
f - g in Lp_Functions (M,k)
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) implies f - g in Lp_Functions (M,k) )
assume A1: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) ; ::_thesis: f - g in Lp_Functions (M,k)
then  (- 1) (#) g in Lp_Functions (M,k) by Th26;
hence  f - g in Lp_Functions (M,k) by Th25, A1; ::_thesis: verum
end;

theorem Th28: :: LPSPACE2:28
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
abs f in Lp_Functions (M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
abs f in Lp_Functions (M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
abs f in Lp_Functions (M,k)
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
abs f in Lp_Functions (M,k)
let f be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) holds
abs f in Lp_Functions (M,k)
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) implies abs f in Lp_Functions (M,k) )
set W = Lp_Functions (M,k);
assume  f in Lp_Functions (M,k) ; ::_thesis: abs f in Lp_Functions (M,k)
then consider f1 being PartFunc of X,REAL such that
A1: ( f1 = f & ex Ef1 being Element of S st
( M . (Ef1 `) = 0 & dom f1 = Ef1 & f1 is_measurable_on Ef1 & (abs f1) to_power k is_integrable_on M ) ) ;
consider Ef being Element of S such that
A2: ( M . (Ef `) = 0 & dom f1 = Ef & f1 is_measurable_on Ef & (abs f1) to_power k is_integrable_on M ) by A1;
 dom (abs f1) = Ef by A2, VALUED_1:def_11;
then  ex Ef being Element of S st
( M . (Ef `) = 0 & dom (abs f1) = Ef & abs f1 is_measurable_on Ef & (abs (abs f1)) to_power k is_integrable_on M ) by A2, MESFUNC6:48;
hence  abs f in Lp_Functions (M,k) by A1; ::_thesis: verum
end;

Lm2:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( Lp_Functions (M,k) is add-closed & Lp_Functions (M,k) is multi-closed )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( Lp_Functions (M,k) is add-closed & Lp_Functions (M,k) is multi-closed )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real holds
( Lp_Functions (M,k) is add-closed & Lp_Functions (M,k) is multi-closed )
let M be sigma_Measure of S; ::_thesis: for k being positive Real holds
( Lp_Functions (M,k) is add-closed & Lp_Functions (M,k) is multi-closed )
let k be positive Real; ::_thesis: ( Lp_Functions (M,k) is add-closed & Lp_Functions (M,k) is multi-closed )
set W = Lp_Functions (M,k);
now__::_thesis:_for_v,_u_being_Element_of_the_carrier_of_(RLSp_PFunct_X)_st_v_in_Lp_Functions_(M,k)_&_u_in_Lp_Functions_(M,k)_holds_
v_+_u_in_Lp_Functions_(M,k)
let v, u be Element of the carrier of (RLSp_PFunct X); ::_thesis: ( v in Lp_Functions (M,k) & u in Lp_Functions (M,k) implies v + u in Lp_Functions (M,k) )
assume A1: ( v in Lp_Functions (M,k) & u in Lp_Functions (M,k) ) ; ::_thesis: v + u in Lp_Functions (M,k)
then consider v1 being PartFunc of X,REAL such that
A2: ( v1 = v & ex ND being Element of S st
( M . (ND `) = 0 & dom v1 = ND & v1 is_measurable_on ND & (abs v1) to_power k is_integrable_on M ) ) ;
consider u1 being PartFunc of X,REAL such that
A3: ( u1 = u & ex ND being Element of S st
( M . (ND `) = 0 & dom u1 = ND & u1 is_measurable_on ND & (abs u1) to_power k is_integrable_on M ) ) by A1;
reconsider h = v + u as Element of PFuncs (X,REAL) ;
 ( dom h = (dom v1) /\ (dom u1) & ( for x being set st x in dom h holds
h . x = (v1 . x) + (u1 . x) ) ) by A2, A3, LPSPACE1:6;
then  v + u = v1 + u1 by VALUED_1:def_1;
hence  v + u in Lp_Functions (M,k) by A1, A2, A3, Th25; ::_thesis: verum
end;
hence  Lp_Functions (M,k) is add-closed by IDEAL_1:def_1; ::_thesis: Lp_Functions (M,k) is multi-closed
now__::_thesis:_for_a_being_Real
for_u_being_VECTOR_of_(RLSp_PFunct_X)_st_u_in_Lp_Functions_(M,k)_holds_
a_*_u_in_Lp_Functions_(M,k)
let a be Real; ::_thesis: for u being VECTOR of (RLSp_PFunct X) st u in Lp_Functions (M,k) holds
a * u in Lp_Functions (M,k)
let u be VECTOR of (RLSp_PFunct X); ::_thesis: ( u in Lp_Functions (M,k) implies a * u in Lp_Functions (M,k) )
assume A4: u in Lp_Functions (M,k) ; ::_thesis: a * u in Lp_Functions (M,k)
then consider u1 being PartFunc of X,REAL such that
A5: ( u1 = u & ex ND being Element of S st
( M . (ND `) = 0 & dom u1 = ND & u1 is_measurable_on ND & (abs u1) to_power k is_integrable_on M ) ) ;
reconsider h = a * u as Element of PFuncs (X,REAL) ;
A6: ( dom h = dom u1 & ( for x being Element of X st x in dom u1 holds
h . x = a * (u1 . x) ) ) by A5, LPSPACE1:9;
then  for x being set st x in dom h holds
h . x = a * (u1 . x) ;
then  a * u = a (#) u1 by A6, VALUED_1:def_5;
hence  a * u in Lp_Functions (M,k) by Th26, A4, A5; ::_thesis: verum
end;
hence  Lp_Functions (M,k) is multi-closed by LPSPACE1:def_1; ::_thesis: verum
end;

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
cluster Lp_Functions (M,k) ->  non empty add-closed multi-closed ;
coherence
 ( Lp_Functions (M,k) is multi-closed & Lp_Functions (M,k) is add-closed ) by Lm2;
end;

Lm3:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is Abelian & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is add-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is right_zeroed & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is vector-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-unital )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is Abelian & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is add-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is right_zeroed & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is vector-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-unital )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real holds
( RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is Abelian & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is add-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is right_zeroed & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is vector-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-unital )
let M be sigma_Measure of S; ::_thesis: for k being positive Real holds
( RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is Abelian & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is add-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is right_zeroed & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is vector-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-unital )
let k be positive Real; ::_thesis: ( RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is Abelian & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is add-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is right_zeroed & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is vector-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-unital )
 0. (RLSp_PFunct X) in Lp_Functions (M,k) by Th23;
hence  ( RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is Abelian & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is add-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is right_zeroed & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is vector-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-unital ) by LPSPACE1:3; ::_thesis: verum
end;

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
cluster RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) ->  Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
coherence
 ( RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is Abelian & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is add-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is right_zeroed & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is vector-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-distributive & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-associative & RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is scalar-unital ) by Lm3;
end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func RLSp_LpFunct (M,k) ->  non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct  equals :: LPSPACE2:def 3
 RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #);
coherence
 RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct ;
end;

:: deftheorem  defines RLSp_LpFunct LPSPACE2:def_3_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds RLSp_LpFunct (M,k) = RLSStruct(# (Lp_Functions (M,k)),(In ((0. (RLSp_PFunct X)),(Lp_Functions (M,k)))),(add| ((Lp_Functions (M,k)),(RLSp_PFunct X))),(Mult_ (Lp_Functions (M,k))) #);

begin


theorem Th29: :: LPSPACE2:29
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for v, u being VECTOR of (RLSp_LpFunct (M,k)) st f = v & g = u holds
f + g = v + u
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for v, u being VECTOR of (RLSp_LpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for v, u being VECTOR of (RLSp_LpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real
for v, u being VECTOR of (RLSp_LpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real
for v, u being VECTOR of (RLSp_LpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let k be positive Real; ::_thesis: for v, u being VECTOR of (RLSp_LpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let v, u be VECTOR of (RLSp_LpFunct (M,k)); ::_thesis: ( f = v & g = u implies f + g = v + u )
reconsider v2 = v, u2 = u as VECTOR of (RLSp_PFunct X) by TARSKI:def_3;
reconsider h = v2 + u2 as Element of PFuncs (X,REAL) ;
reconsider v2 = v2, u2 = u2 as Element of PFuncs (X,REAL) ;
assume A1: ( f = v & g = u ) ; ::_thesis: f + g = v + u
A2: ( dom h = (dom v2) /\ (dom u2) & ( for x being Element of X st x in dom h holds
h . x = (v2 . x) + (u2 . x) ) ) by LPSPACE1:6;
 for x being set st x in dom h holds
h . x = (f . x) + (g . x) by A1, LPSPACE1:6;
then  h = f + g by A1, A2, VALUED_1:def_1;
hence  f + g = v + u by LPSPACE1:4; ::_thesis: verum
end;

theorem Th30: :: LPSPACE2:30
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let f be PartFunc of X,REAL; ::_thesis: for a being Real
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let a be Real; ::_thesis: for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let k be positive Real; ::_thesis: for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let u be VECTOR of (RLSp_LpFunct (M,k)); ::_thesis: ( f = u implies a (#) f = a * u )
reconsider u2 = u as VECTOR of (RLSp_PFunct X) by TARSKI:def_3;
reconsider h = a * u2 as Element of PFuncs (X,REAL) ;
assume A1: f = u ; ::_thesis: a (#) f = a * u
then A2: dom h = dom f by LPSPACE1:9;
then  for x being set st x in dom h holds
h . x = a * (f . x) by A1, LPSPACE1:9;
then  h = a (#) f by A2, VALUED_1:def_5;
hence  a (#) f = a * u by LPSPACE1:5; ::_thesis: verum
end;

theorem Th31: :: LPSPACE2:31
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
( u + ((- 1) * u) = (X --> 0) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
( u + ((- 1) * u) = (X --> 0) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
( u + ((- 1) * u) = (X --> 0) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
( u + ((- 1) * u) = (X --> 0) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
let f be PartFunc of X,REAL; ::_thesis: for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
( u + ((- 1) * u) = (X --> 0) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
let k be positive Real; ::_thesis: for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
( u + ((- 1) * u) = (X --> 0) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
let u be VECTOR of (RLSp_LpFunct (M,k)); ::_thesis: ( f = u implies ( u + ((- 1) * u) = (X --> 0) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) ) )
reconsider u2 = u as VECTOR of (RLSp_PFunct X) by TARSKI:def_3;
assume A1: f = u ; ::_thesis: ( u + ((- 1) * u) = (X --> 0) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
 (- 1) * u = (- 1) * u2 by LPSPACE1:5;
then A2: u + ((- 1) * u) = u2 + ((- 1) * u2) by LPSPACE1:4;
hence  u + ((- 1) * u) = (X --> 0) | (dom f) by A1, LPSPACE1:16; ::_thesis: ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M )
 u + ((- 1) * u) in Lp_Functions (M,k) ;
then consider v being PartFunc of X,REAL such that
A3: ( v = u + ((- 1) * u) & ex ND being Element of S st
( M . (ND `) = 0 & dom v = ND & v is_measurable_on ND & (abs v) to_power k is_integrable_on M ) ) ;
 u in Lp_Functions (M,k) ;
then  ex uu1 being PartFunc of X,REAL st
( uu1 = u & ex ND being Element of S st
( M . (ND `) = 0 & dom uu1 = ND & uu1 is_measurable_on ND & (abs uu1) to_power k is_integrable_on M ) ) ;
then consider ND being Element of S such that
A4: ( M . (ND `) = 0 & dom f = ND & f is_measurable_on ND & (abs f) to_power k is_integrable_on M ) by A1;
set g = X --> 0;
A5: ( ND ` is Element of S & (ND `) ` = ND ) by MEASURE1:34;
A6: X --> 0 in Lp_Functions (M,k) by Th23;
 v | ND = ((X --> 0) | ND) | ND by A2, A3, A4, A1, LPSPACE1:16;
then  v | ND = (X --> 0) | ND by FUNCT_1:51;
then  v a.e.= X --> 0,M by A4, A5, LPSPACE1:def_10;
hence  ex v, g being PartFunc of X,REAL st
( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) by A3, A6; ::_thesis: verum
end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func AlmostZeroLpFunctions (M,k) ->  non empty Subset of (RLSp_LpFunct (M,k)) equals :: LPSPACE2:def 4
 {  f where f is PartFunc of X,REAL : ( f in Lp_Functions (M,k) & f a.e.= X --> 0,M )  }  ;
coherence
  {  f where f is PartFunc of X,REAL : ( f in Lp_Functions (M,k) & f a.e.= X --> 0,M )  }  is non empty Subset of (RLSp_LpFunct (M,k))
proof
A1: now__::_thesis:_for_x_being_set_st_x_in__{__f_where_f_is_PartFunc_of_X,REAL_:_(_f_in_Lp_Functions_(M,k)_&_f_a.e.=_X_-->_0,M_)__}__holds_
x_in_the_carrier_of_(RLSp_LpFunct_(M,k))
let x be set ; ::_thesis: ( x in  {  f where f is PartFunc of X,REAL : ( f in Lp_Functions (M,k) & f a.e.= X --> 0,M )  }  implies x in the carrier of (RLSp_LpFunct (M,k)) )
assume  x in  {  f where f is PartFunc of X,REAL : ( f in Lp_Functions (M,k) & f a.e.= X --> 0,M )  }  ; ::_thesis: x in the carrier of (RLSp_LpFunct (M,k))
then  ex f being PartFunc of X,REAL st
( x = f & f in Lp_Functions (M,k) & f a.e.= X --> 0,M ) ;
hence  x in the carrier of (RLSp_LpFunct (M,k)) ; ::_thesis: verum
end;
A2: X --> 0 a.e.= X --> 0,M by LPSPACE1:28;
 X --> 0 in Lp_Functions (M,k) by Th23;
then  X --> 0 in  {  f where f is PartFunc of X,REAL : ( f in Lp_Functions (M,k) & f a.e.= X --> 0,M )  }  by A2;
hence   {  f where f is PartFunc of X,REAL : ( f in Lp_Functions (M,k) & f a.e.= X --> 0,M )  }  is non empty Subset of (RLSp_LpFunct (M,k)) by A1, TARSKI:def_3; ::_thesis: verum
end;
end;

:: deftheorem  defines AlmostZeroLpFunctions LPSPACE2:def_4_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds AlmostZeroLpFunctions (M,k) =  {  f where f is PartFunc of X,REAL : ( f in Lp_Functions (M,k) & f a.e.= X --> 0,M )  }  ;

Lm4:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( AlmostZeroLpFunctions (M,k) is add-closed & AlmostZeroLpFunctions (M,k) is multi-closed )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( AlmostZeroLpFunctions (M,k) is add-closed & AlmostZeroLpFunctions (M,k) is multi-closed )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real holds
( AlmostZeroLpFunctions (M,k) is add-closed & AlmostZeroLpFunctions (M,k) is multi-closed )
let M be sigma_Measure of S; ::_thesis: for k being positive Real holds
( AlmostZeroLpFunctions (M,k) is add-closed & AlmostZeroLpFunctions (M,k) is multi-closed )
let k be positive Real; ::_thesis: ( AlmostZeroLpFunctions (M,k) is add-closed & AlmostZeroLpFunctions (M,k) is multi-closed )
set Z = AlmostZeroLpFunctions (M,k);
set V = RLSp_LpFunct (M,k);
now__::_thesis:_for_v,_u_being_VECTOR_of_(RLSp_LpFunct_(M,k))_st_v_in_AlmostZeroLpFunctions_(M,k)_&_u_in_AlmostZeroLpFunctions_(M,k)_holds_
v_+_u_in_AlmostZeroLpFunctions_(M,k)
let v, u be VECTOR of (RLSp_LpFunct (M,k)); ::_thesis: ( v in AlmostZeroLpFunctions (M,k) & u in AlmostZeroLpFunctions (M,k) implies v + u in AlmostZeroLpFunctions (M,k) )
assume A1: ( v in AlmostZeroLpFunctions (M,k) & u in AlmostZeroLpFunctions (M,k) ) ; ::_thesis: v + u in AlmostZeroLpFunctions (M,k)
then consider v1 being PartFunc of X,REAL such that
A2: ( v1 = v & v1 in Lp_Functions (M,k) & v1 a.e.= X --> 0,M ) ;
consider u1 being PartFunc of X,REAL such that
A3: ( u1 = u & u1 in Lp_Functions (M,k) & u1 a.e.= X --> 0,M ) by A1;
A4: v + u = v1 + u1 by Th29, A2, A3;
 (X --> 0) + (X --> 0) = X --> 0 by LPSPACE1:33;
then  ( v1 + u1 in Lp_Functions (M,k) & v1 + u1 a.e.= X --> 0,M ) by A4, A2, A3, LPSPACE1:31;
hence  v + u in AlmostZeroLpFunctions (M,k) by A4; ::_thesis: verum
end;
hence  AlmostZeroLpFunctions (M,k) is add-closed by IDEAL_1:def_1; ::_thesis: AlmostZeroLpFunctions (M,k) is multi-closed
now__::_thesis:_for_a_being_Real
for_u_being_VECTOR_of_(RLSp_LpFunct_(M,k))_st_u_in_AlmostZeroLpFunctions_(M,k)_holds_
a_*_u_in_AlmostZeroLpFunctions_(M,k)
let a be Real; ::_thesis: for u being VECTOR of (RLSp_LpFunct (M,k)) st u in AlmostZeroLpFunctions (M,k) holds
a * u in AlmostZeroLpFunctions (M,k)
let u be VECTOR of (RLSp_LpFunct (M,k)); ::_thesis: ( u in AlmostZeroLpFunctions (M,k) implies a * u in AlmostZeroLpFunctions (M,k) )
assume  u in AlmostZeroLpFunctions (M,k) ; ::_thesis: a * u in AlmostZeroLpFunctions (M,k)
then consider u1 being PartFunc of X,REAL such that
A5: ( u1 = u & u1 in Lp_Functions (M,k) & u1 a.e.= X --> 0,M ) ;
A6: a * u = a (#) u1 by Th30, A5;
 a (#) (X --> 0) = X --> 0 by LPSPACE1:33;
then  ( a (#) u1 in Lp_Functions (M,k) & a (#) u1 a.e.= X --> 0,M ) by A6, A5, LPSPACE1:32;
hence  a * u in AlmostZeroLpFunctions (M,k) by A6; ::_thesis: verum
end;
hence  AlmostZeroLpFunctions (M,k) is multi-closed by LPSPACE1:def_1; ::_thesis: verum
end;

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
cluster AlmostZeroLpFunctions (M,k) ->  non empty add-closed multi-closed ;
coherence
 ( AlmostZeroLpFunctions (M,k) is add-closed & AlmostZeroLpFunctions (M,k) is multi-closed ) by Lm4;
end;

theorem Th32: :: LPSPACE2:32
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( 0. (RLSp_LpFunct (M,k)) = X --> 0 & 0. (RLSp_LpFunct (M,k)) in AlmostZeroLpFunctions (M,k) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( 0. (RLSp_LpFunct (M,k)) = X --> 0 & 0. (RLSp_LpFunct (M,k)) in AlmostZeroLpFunctions (M,k) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real holds
( 0. (RLSp_LpFunct (M,k)) = X --> 0 & 0. (RLSp_LpFunct (M,k)) in AlmostZeroLpFunctions (M,k) )
let M be sigma_Measure of S; ::_thesis: for k being positive Real holds
( 0. (RLSp_LpFunct (M,k)) = X --> 0 & 0. (RLSp_LpFunct (M,k)) in AlmostZeroLpFunctions (M,k) )
let k be positive Real; ::_thesis: ( 0. (RLSp_LpFunct (M,k)) = X --> 0 & 0. (RLSp_LpFunct (M,k)) in AlmostZeroLpFunctions (M,k) )
thus  0. (RLSp_LpFunct (M,k)) = X --> 0 by Th23, FUNCT_7:def_1; ::_thesis: 0. (RLSp_LpFunct (M,k)) in AlmostZeroLpFunctions (M,k)
A1: ( X --> 0 a.e.= X --> 0,M & X --> 0 in Lp_Functions (M,k) ) by Th23, LPSPACE1:28;
 0. (RLSp_LpFunct (M,k)) = 0. (RLSp_PFunct X) by Th23, FUNCT_7:def_1;
hence  0. (RLSp_LpFunct (M,k)) in AlmostZeroLpFunctions (M,k) by A1; ::_thesis: verum
end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func RLSp_AlmostZeroLpFunct (M,k) ->  non empty RLSStruct  equals :: LPSPACE2:def 5
 RLSStruct(# (AlmostZeroLpFunctions (M,k)),(In ((0. (RLSp_LpFunct (M,k))),(AlmostZeroLpFunctions (M,k)))),(add| ((AlmostZeroLpFunctions (M,k)),(RLSp_LpFunct (M,k)))),(Mult_ (AlmostZeroLpFunctions (M,k))) #);
coherence
 RLSStruct(# (AlmostZeroLpFunctions (M,k)),(In ((0. (RLSp_LpFunct (M,k))),(AlmostZeroLpFunctions (M,k)))),(add| ((AlmostZeroLpFunctions (M,k)),(RLSp_LpFunct (M,k)))),(Mult_ (AlmostZeroLpFunctions (M,k))) #) is non empty RLSStruct ;
end;

:: deftheorem  defines RLSp_AlmostZeroLpFunct LPSPACE2:def_5_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds RLSp_AlmostZeroLpFunct (M,k) = RLSStruct(# (AlmostZeroLpFunctions (M,k)),(In ((0. (RLSp_LpFunct (M,k))),(AlmostZeroLpFunctions (M,k)))),(add| ((AlmostZeroLpFunctions (M,k)),(RLSp_LpFunct (M,k)))),(Mult_ (AlmostZeroLpFunctions (M,k))) #);

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
cluster RLSp_LpFunct (M,k) ->  non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
coherence
 ( RLSp_LpFunct (M,k) is strict & RLSp_LpFunct (M,k) is Abelian & RLSp_LpFunct (M,k) is add-associative & RLSp_LpFunct (M,k) is right_zeroed & RLSp_LpFunct (M,k) is vector-distributive & RLSp_LpFunct (M,k) is scalar-distributive & RLSp_LpFunct (M,k) is scalar-associative & RLSp_LpFunct (M,k) is scalar-unital ) ;
end;


theorem :: LPSPACE2:33
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for v, u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = v & g = u holds
f + g = v + u
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for v, u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for v, u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real
for v, u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real
for v, u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let k be positive Real; ::_thesis: for v, u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = v & g = u holds
f + g = v + u
let v, u be VECTOR of (RLSp_AlmostZeroLpFunct (M,k)); ::_thesis: ( f = v & g = u implies f + g = v + u )
reconsider v2 = v, u2 = u as VECTOR of (RLSp_LpFunct (M,k)) by TARSKI:def_3;
assume A1: ( f = v & g = u ) ; ::_thesis: f + g = v + u
 v + u = v2 + u2 by LPSPACE1:4;
hence  v + u = f + g by Th29, A1; ::_thesis: verum
end;

theorem :: LPSPACE2:34
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = u holds
a (#) f = a * u
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = u holds
a (#) f = a * u
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = u holds
a (#) f = a * u
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = u holds
a (#) f = a * u
let f be PartFunc of X,REAL; ::_thesis: for a being Real
for k being positive Real
for u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = u holds
a (#) f = a * u
let a be Real; ::_thesis: for k being positive Real
for u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = u holds
a (#) f = a * u
let k be positive Real; ::_thesis: for u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = u holds
a (#) f = a * u
let u be VECTOR of (RLSp_AlmostZeroLpFunct (M,k)); ::_thesis: ( f = u implies a (#) f = a * u )
reconsider u2 = u as VECTOR of (RLSp_LpFunct (M,k)) by TARSKI:def_3;
assume A1: f = u ; ::_thesis: a (#) f = a * u
 a * u = a * u2 by LPSPACE1:5;
hence  a * u = a (#) f by Th30, A1; ::_thesis: verum
end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let f be PartFunc of X,REAL;
let k be positive Real;
func a.e-eq-class_Lp (f,M,k) ->  Subset of (Lp_Functions (M,k)) equals :: LPSPACE2:def 6
 {  h where h is PartFunc of X,REAL : ( h in Lp_Functions (M,k) & f a.e.= h,M )  }  ;
correctness
coherence
{  h where h is PartFunc of X,REAL : ( h in Lp_Functions (M,k) & f a.e.= h,M )  }  is Subset of (Lp_Functions (M,k));
proof
now__::_thesis:_for_x_being_set_st_x_in__{__g_where_g_is_PartFunc_of_X,REAL_:_(_g_in_Lp_Functions_(M,k)_&_f_a.e.=_g,M_)__}__holds_
x_in_Lp_Functions_(M,k)
let x be set ; ::_thesis: ( x in  {  g where g is PartFunc of X,REAL : ( g in Lp_Functions (M,k) & f a.e.= g,M )  }  implies x in Lp_Functions (M,k) )
assume  x in  {  g where g is PartFunc of X,REAL : ( g in Lp_Functions (M,k) & f a.e.= g,M )  }  ; ::_thesis: x in Lp_Functions (M,k)
then  ex g being PartFunc of X,REAL st
( x = g & g in Lp_Functions (M,k) & f a.e.= g,M ) ;
hence  x in Lp_Functions (M,k) ; ::_thesis: verum
end;
hence   {  h where h is PartFunc of X,REAL : ( h in Lp_Functions (M,k) & f a.e.= h,M )  }  is Subset of (Lp_Functions (M,k)) by TARSKI:def_3; ::_thesis: verum
end;
end;

:: deftheorem  defines a.e-eq-class_Lp LPSPACE2:def_6_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real holds a.e-eq-class_Lp (f,M,k) =  {  h where h is PartFunc of X,REAL : ( h in Lp_Functions (M,k) & f a.e.= h,M )  }  ;

theorem Th35: :: LPSPACE2:35
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E )
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E )
let f be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) holds
ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E )
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) implies ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) )
assume  f in Lp_Functions (M,k) ; ::_thesis: ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E )
then  ex f1 being PartFunc of X,REAL st
( f = f1 & ex E being Element of S st
( M . (E `) = 0 & dom f1 = E & f1 is_measurable_on E & (abs f1) to_power k is_integrable_on M ) ) ;
hence  ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) ; ::_thesis: verum
end;

theorem Th36: :: LPSPACE2:36
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for g, f being PartFunc of X,REAL
for k being positive Real st g in Lp_Functions (M,k) & g a.e.= f,M holds
g in a.e-eq-class_Lp (f,M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for g, f being PartFunc of X,REAL
for k being positive Real st g in Lp_Functions (M,k) & g a.e.= f,M holds
g in a.e-eq-class_Lp (f,M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for g, f being PartFunc of X,REAL
for k being positive Real st g in Lp_Functions (M,k) & g a.e.= f,M holds
g in a.e-eq-class_Lp (f,M,k)
let M be sigma_Measure of S; ::_thesis: for g, f being PartFunc of X,REAL
for k being positive Real st g in Lp_Functions (M,k) & g a.e.= f,M holds
g in a.e-eq-class_Lp (f,M,k)
let g, f be PartFunc of X,REAL; ::_thesis: for k being positive Real st g in Lp_Functions (M,k) & g a.e.= f,M holds
g in a.e-eq-class_Lp (f,M,k)
let k be positive Real; ::_thesis: ( g in Lp_Functions (M,k) & g a.e.= f,M implies g in a.e-eq-class_Lp (f,M,k) )
assume that
A1: g in Lp_Functions (M,k) and
A2: g a.e.= f,M ; ::_thesis: g in a.e-eq-class_Lp (f,M,k)
 f a.e.= g,M by A2, LPSPACE1:29;
hence  g in a.e-eq-class_Lp (f,M,k) by A1; ::_thesis: verum
end;

theorem Th37: :: LPSPACE2:37
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & g in a.e-eq-class_Lp (f,M,k) holds
( g a.e.= f,M & f in Lp_Functions (M,k) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & g in a.e-eq-class_Lp (f,M,k) holds
( g a.e.= f,M & f in Lp_Functions (M,k) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & g in a.e-eq-class_Lp (f,M,k) holds
( g a.e.= f,M & f in Lp_Functions (M,k) )
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & g in a.e-eq-class_Lp (f,M,k) holds
( g a.e.= f,M & f in Lp_Functions (M,k) )
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & g in a.e-eq-class_Lp (f,M,k) holds
( g a.e.= f,M & f in Lp_Functions (M,k) )
let k be positive Real; ::_thesis: ( ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & g in a.e-eq-class_Lp (f,M,k) implies ( g a.e.= f,M & f in Lp_Functions (M,k) ) )
assume that
A1: ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) and
A2: g in a.e-eq-class_Lp (f,M,k) ; ::_thesis: ( g a.e.= f,M & f in Lp_Functions (M,k) )
A3: ex r being PartFunc of X,REAL st
( g = r & r in Lp_Functions (M,k) & f a.e.= r,M ) by A2;
hence  g a.e.= f,M by LPSPACE1:29; ::_thesis: f in Lp_Functions (M,k)
 g in Lp_Functions (M,k) by A2;
then consider g1 being PartFunc of X,REAL such that
A4: ( g = g1 & ex E being Element of S st
( M . (E `) = 0 & dom g1 = E & g1 is_measurable_on E & (abs g1) to_power k is_integrable_on M ) ) ;
consider Eh being Element of S such that
A5: ( M . (Eh `) = 0 & dom g = Eh & g is_measurable_on Eh & (abs g) to_power k is_integrable_on M ) by A4;
reconsider ND = Eh ` as Element of S by MEASURE1:34;
 ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E & (abs f) to_power k is_integrable_on M )
proof
set AFK = (abs f) to_power k;
set AGK = (abs g) to_power k;
consider Ef being Element of S such that
A6: ( M . (Ef `) = 0 & Ef = dom f & f is_measurable_on Ef ) by A1;
take  Ef ; ::_thesis: ( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M )
consider EE being Element of S such that
A7: ( M . EE = 0 & g | (EE `) = f | (EE `) ) by A3, LPSPACE1:def_10;
reconsider E1 = ND \/ EE as Element of S ;
 EE c= E1 by XBOOLE_1:7;
then  E1 ` c= EE ` by SUBSET_1:12;
then A8: ( f | (E1 `) = (f | (EE `)) | (E1 `) & g | (E1 `) = (g | (EE `)) | (E1 `) ) by FUNCT_1:51;
A9: dom (abs f) = Ef by A6, VALUED_1:def_11;
then  dom ((abs f) to_power k) = Ef by MESFUN6C:def_4;
then A10: ( dom (max+ (R_EAL ((abs f) to_power k))) = Ef & dom (max- (R_EAL ((abs f) to_power k))) = Ef ) by MESFUNC2:def_2, MESFUNC2:def_3;
 abs f is_measurable_on Ef by A6, MESFUNC6:48;
then  (abs f) to_power k is_measurable_on Ef by A9, MESFUN6C:29;
then A11: ( Ef = dom (R_EAL ((abs f) to_power k)) & R_EAL ((abs f) to_power k) is_measurable_on Ef ) by A9, MESFUN6C:def_4, MESFUNC6:def_1;
then A12: ( max+ (R_EAL ((abs f) to_power k)) is_measurable_on Ef & max- (R_EAL ((abs f) to_power k)) is_measurable_on Ef ) by MESFUNC2:25, MESFUNC2:26;
 ( ( for x being Element of X holds 0. <= (max+ (R_EAL ((abs f) to_power k))) . x ) & ( for x being Element of X holds 0. <= (max- (R_EAL ((abs f) to_power k))) . x ) ) by MESFUNC2:12, MESFUNC2:13;
then A13: ( max+ (R_EAL ((abs f) to_power k)) is nonnegative & max- (R_EAL ((abs f) to_power k)) is nonnegative ) by SUPINF_2:39;
A14: Ef = (Ef /\ E1) \/ (Ef \ E1) by XBOOLE_1:51;
reconsider E0 = Ef /\ E1 as Element of S ;
reconsider E2 = Ef \ E1 as Element of S ;
 ( max+ (R_EAL ((abs f) to_power k)) = (max+ (R_EAL ((abs f) to_power k))) | (dom (max+ (R_EAL ((abs f) to_power k)))) & max- (R_EAL ((abs f) to_power k)) = (max- (R_EAL ((abs f) to_power k))) | (dom (max- (R_EAL ((abs f) to_power k)))) ) by RELAT_1:69;
then A15: ( integral+ (M,(max+ (R_EAL ((abs f) to_power k)))) = (integral+ (M,((max+ (R_EAL ((abs f) to_power k))) | E0))) + (integral+ (M,((max+ (R_EAL ((abs f) to_power k))) | E2))) & integral+ (M,(max- (R_EAL ((abs f) to_power k)))) = (integral+ (M,((max- (R_EAL ((abs f) to_power k))) | E0))) + (integral+ (M,((max- (R_EAL ((abs f) to_power k))) | E2))) ) by A10, A12, A13, A14, MESFUNC5:81, XBOOLE_1:89;
A16: ( integral+ (M,((max+ (R_EAL ((abs f) to_power k))) | E0)) >= 0 & integral+ (M,((max- (R_EAL ((abs f) to_power k))) | E0)) >= 0 ) by A12, A13, A10, MESFUNC5:80;
 ( ND is measure_zero of M & EE is measure_zero of M ) by A5, A7, MEASURE1:def_7;
then  E1 is measure_zero of M by MEASURE1:37;
then  M . E1 = 0 by MEASURE1:def_7;
then  ( integral+ (M,((max+ (R_EAL ((abs f) to_power k))) | E1)) = 0 & integral+ (M,((max- (R_EAL ((abs f) to_power k))) | E1)) = 0 ) by A10, A12, A13, MESFUNC5:82;
then  ( integral+ (M,((max+ (R_EAL ((abs f) to_power k))) | E0)) = 0 & integral+ (M,((max- (R_EAL ((abs f) to_power k))) | E0)) = 0 ) by A10, A12, A13, A16, MESFUNC5:83, XBOOLE_1:17;
then A17: ( integral+ (M,(max+ (R_EAL ((abs f) to_power k)))) = integral+ (M,((max+ (R_EAL ((abs f) to_power k))) | E2)) & integral+ (M,(max- (R_EAL ((abs f) to_power k)))) = integral+ (M,((max- (R_EAL ((abs f) to_power k))) | E2)) ) by A15, XXREAL_3:4;
 Ef \ E1 = Ef /\ (E1 `) by SUBSET_1:13;
then A18: E2 c= E1 ` by XBOOLE_1:17;
then  f | E2 = (g | (E1 `)) | E2 by A7, A8, FUNCT_1:51;
then A19: f | E2 = g | E2 by A18, FUNCT_1:51;
A20: ( (abs f) | E2 = abs (f | E2) & (abs g) | E2 = abs (g | E2) ) by RFUNCT_1:46;
A21: ( ((abs f) | E2) to_power k = ((abs f) to_power k) | E2 & ((abs g) | E2) to_power k = ((abs g) to_power k) | E2 ) by Th20;
A22: ( (max+ (R_EAL ((abs f) to_power k))) | E2 = max+ ((R_EAL ((abs f) to_power k)) | E2) & (max+ (R_EAL ((abs g) to_power k))) | E2 = max+ ((R_EAL ((abs g) to_power k)) | E2) & (max- (R_EAL ((abs f) to_power k))) | E2 = max- ((R_EAL ((abs f) to_power k)) | E2) & (max- (R_EAL ((abs g) to_power k))) | E2 = max- ((R_EAL ((abs g) to_power k)) | E2) ) by MESFUNC5:28;
A23: R_EAL ((abs g) to_power k) is_integrable_on M by A5, MESFUNC6:def_4;
then A24: ( integral+ (M,(max+ (R_EAL ((abs g) to_power k)))) < +infty & integral+ (M,(max- (R_EAL ((abs g) to_power k)))) < +infty ) by MESFUNC5:def_17;
 ( integral+ (M,(max+ ((R_EAL ((abs g) to_power k)) | E2))) <= integral+ (M,(max+ (R_EAL ((abs g) to_power k)))) & integral+ (M,(max- ((R_EAL ((abs g) to_power k)) | E2))) <= integral+ (M,(max- (R_EAL ((abs g) to_power k)))) ) by A23, MESFUNC5:97;
then  ( integral+ (M,(max+ (R_EAL ((abs f) to_power k)))) < +infty & integral+ (M,(max- (R_EAL ((abs f) to_power k)))) < +infty ) by A17, A19, A20, A21, A22, A24, XXREAL_0:2;
then  R_EAL ((abs f) to_power k) is_integrable_on M by A11, MESFUNC5:def_17;
hence  ( M . (Ef `) = 0 & dom f = Ef & f is_measurable_on Ef & (abs f) to_power k is_integrable_on M ) by A6, MESFUNC6:def_4; ::_thesis: verum
end;
hence  f in Lp_Functions (M,k) ; ::_thesis: verum
end;

theorem Th38: :: LPSPACE2:38
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
f in a.e-eq-class_Lp (f,M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
f in a.e-eq-class_Lp (f,M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
f in a.e-eq-class_Lp (f,M,k)
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
f in a.e-eq-class_Lp (f,M,k)
let f be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) holds
f in a.e-eq-class_Lp (f,M,k)
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) implies f in a.e-eq-class_Lp (f,M,k) )
assume A1: f in Lp_Functions (M,k) ; ::_thesis: f in a.e-eq-class_Lp (f,M,k)
 f a.e.= f,M by LPSPACE1:28;
hence  f in a.e-eq-class_Lp (f,M,k) by A1; ::_thesis: verum
end;

theorem Th39: :: LPSPACE2:39
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for g, f being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for g, f being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for g, f being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let M be sigma_Measure of S; ::_thesis: for g, f being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let g, f be PartFunc of X,REAL; ::_thesis: for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let k be positive Real; ::_thesis: ( ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) implies f a.e.= g,M )
assume that
A1: ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) and
A2: a.e-eq-class_Lp (f,M,k) <> {} and
A3: a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) ; ::_thesis: f a.e.= g,M
consider x being set  such that
A4: x in a.e-eq-class_Lp (f,M,k) by A2, XBOOLE_0:def_1;
consider r being PartFunc of X,REAL such that
A5: ( x = r & r in Lp_Functions (M,k) & f a.e.= r,M ) by A4;
 r a.e.= g,M by A1, A3, A4, A5, Th37;
hence  f a.e.= g,M by A5, LPSPACE1:30; ::_thesis: verum
end;

theorem :: LPSPACE2:40
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) implies f a.e.= g,M )
assume that
A1: f in Lp_Functions (M,k) and
A2: ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) and
A3: a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) ; ::_thesis: f a.e.= g,M
 not a.e-eq-class_Lp (f,M,k) is empty by A1, Th38;
hence  f a.e.= g,M by A2, A3, Th39; ::_thesis: verum
end;

theorem Th41: :: LPSPACE2:41
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let k be positive Real; ::_thesis: ( f a.e.= g,M implies a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) )
assume A1: f a.e.= g,M ; ::_thesis: a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
now__::_thesis:_for_x_being_set_st_x_in_a.e-eq-class_Lp_(f,M,k)_holds_
x_in_a.e-eq-class_Lp_(g,M,k)
let x be set ; ::_thesis: ( x in a.e-eq-class_Lp (f,M,k) implies x in a.e-eq-class_Lp (g,M,k) )
assume  x in a.e-eq-class_Lp (f,M,k) ; ::_thesis: x in a.e-eq-class_Lp (g,M,k)
then consider r being PartFunc of X,REAL such that
A2: ( x = r & r in Lp_Functions (M,k) & f a.e.= r,M ) ;
 r a.e.= f,M by A2, LPSPACE1:29;
then  r a.e.= g,M by A1, LPSPACE1:30;
then  g a.e.= r,M by LPSPACE1:29;
hence  x in a.e-eq-class_Lp (g,M,k) by A2; ::_thesis: verum
end;
then A3: a.e-eq-class_Lp (f,M,k) c= a.e-eq-class_Lp (g,M,k) by TARSKI:def_3;
now__::_thesis:_for_x_being_set_st_x_in_a.e-eq-class_Lp_(g,M,k)_holds_
x_in_a.e-eq-class_Lp_(f,M,k)
let x be set ; ::_thesis: ( x in a.e-eq-class_Lp (g,M,k) implies x in a.e-eq-class_Lp (f,M,k) )
assume  x in a.e-eq-class_Lp (g,M,k) ; ::_thesis: x in a.e-eq-class_Lp (f,M,k)
then consider r being PartFunc of X,REAL such that
A4: ( x = r & r in Lp_Functions (M,k) & g a.e.= r,M ) ;
 ( r a.e.= g,M & g a.e.= f,M ) by A1, A4, LPSPACE1:29;
then  r a.e.= f,M by LPSPACE1:30;
then  f a.e.= r,M by LPSPACE1:29;
hence  x in a.e-eq-class_Lp (f,M,k) by A4; ::_thesis: verum
end;
then  a.e-eq-class_Lp (g,M,k) c= a.e-eq-class_Lp (f,M,k) by TARSKI:def_3;
hence  a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) by A3, XBOOLE_0:def_10; ::_thesis: verum
end;

theorem Th42: :: LPSPACE2:42
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) by Th41;

theorem :: LPSPACE2:43
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in a.e-eq-class_Lp (f,M,k) holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in a.e-eq-class_Lp (f,M,k) holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in a.e-eq-class_Lp (f,M,k) holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in a.e-eq-class_Lp (f,M,k) holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) & g in a.e-eq-class_Lp (f,M,k) holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) & g in a.e-eq-class_Lp (f,M,k) implies a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) )
assume that
A1: f in Lp_Functions (M,k) and
A2: g in a.e-eq-class_Lp (f,M,k) ; ::_thesis: a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
 ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) by A1, Th35;
hence  a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) by Th41, A2, Th37; ::_thesis: verum
end;

theorem :: LPSPACE2:44
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
let M be sigma_Measure of S; ::_thesis: for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
let f, f1, g, g1 be PartFunc of X,REAL; ::_thesis: for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
let k be positive Real; ::_thesis: ( ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) implies a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k) )
assume  ( ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) ) ; ::_thesis: a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
then  ( f a.e.= f1,M & g a.e.= g1,M ) by Th39;
hence  a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k) by Th41, LPSPACE1:31; ::_thesis: verum
end;

theorem Th45: :: LPSPACE2:45
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & f1 in Lp_Functions (M,k) & g in Lp_Functions (M,k) & g1 in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & f1 in Lp_Functions (M,k) & g in Lp_Functions (M,k) & g1 in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & f1 in Lp_Functions (M,k) & g in Lp_Functions (M,k) & g1 in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
let M be sigma_Measure of S; ::_thesis: for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & f1 in Lp_Functions (M,k) & g in Lp_Functions (M,k) & g1 in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
let f, f1, g, g1 be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) & f1 in Lp_Functions (M,k) & g in Lp_Functions (M,k) & g1 in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) holds
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) & f1 in Lp_Functions (M,k) & g in Lp_Functions (M,k) & g1 in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) implies a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k) )
assume that
A1: f in Lp_Functions (M,k) and
A2: f1 in Lp_Functions (M,k) and
A3: g in Lp_Functions (M,k) and
A4: g1 in Lp_Functions (M,k) and
A5: ( a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (f1,M,k) & a.e-eq-class_Lp (g,M,k) = a.e-eq-class_Lp (g1,M,k) ) ; ::_thesis: a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
A6: ( ex E being Element of S st
( M . (E `) = 0 & dom f1 = E & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g1 = E & g1 is_measurable_on E ) ) by A2, A4, Th35;
 ( f in a.e-eq-class_Lp (f,M,k) & g in a.e-eq-class_Lp (g,M,k) ) by A1, A3, Th38;
then  ( f a.e.= f1,M & g a.e.= g1,M ) by A5, A6, Th37;
hence  a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k) by Th41, LPSPACE1:31; ::_thesis: verum
end;

theorem :: LPSPACE2:46
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let f, g be PartFunc of X,REAL; ::_thesis: for a being Real
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let a be Real; ::_thesis: for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let k be positive Real; ::_thesis: ( ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) implies a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k) )
assume  ( ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) ) ; ::_thesis: a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
then  a (#) f a.e.= a (#) g,M by Th39, LPSPACE1:32;
hence  a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k) by Th41; ::_thesis: verum
end;

theorem Th47: :: LPSPACE2:47
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let f, g be PartFunc of X,REAL; ::_thesis: for a being Real
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let a be Real; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) implies a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k) )
assume A1: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) ) ; ::_thesis: a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)
then A2: ( ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E ) ) by Th35;
 f in a.e-eq-class_Lp (g,M,k) by A1, Th38;
then  ( f a.e.= g,M & a (#) f in Lp_Functions (M,k) & a (#) g in Lp_Functions (M,k) ) by A2, Th37, Th26;
hence  a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k) by Th42, LPSPACE1:32; ::_thesis: verum
end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func CosetSet (M,k) ->  non empty Subset-Family of (Lp_Functions (M,k)) equals :: LPSPACE2:def 7
 {  (a.e-eq-class_Lp (f,M,k)) where f is PartFunc of X,REAL : f in Lp_Functions (M,k)  }  ;
correctness
coherence
{  (a.e-eq-class_Lp (f,M,k)) where f is PartFunc of X,REAL : f in Lp_Functions (M,k)  }  is non empty Subset-Family of (Lp_Functions (M,k));
proof
set C =  {  (a.e-eq-class_Lp (f,M,k)) where f is PartFunc of X,REAL : f in Lp_Functions (M,k)  }  ;
A1:  {  (a.e-eq-class_Lp (f,M,k)) where f is PartFunc of X,REAL : f in Lp_Functions (M,k)  }  c= bool (Lp_Functions (M,k))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  (a.e-eq-class_Lp (f,M,k)) where f is PartFunc of X,REAL : f in Lp_Functions (M,k)  }  or x in bool (Lp_Functions (M,k)) )
assume  x in  {  (a.e-eq-class_Lp (f,M,k)) where f is PartFunc of X,REAL : f in Lp_Functions (M,k)  }  ; ::_thesis: x in bool (Lp_Functions (M,k))
then  ex f being PartFunc of X,REAL st
( a.e-eq-class_Lp (f,M,k) = x & f in Lp_Functions (M,k) ) ;
hence  x in bool (Lp_Functions (M,k)) ; ::_thesis: verum
end;
 X --> 0 in Lp_Functions (M,k) by Th23;
then  a.e-eq-class_Lp ((X --> 0),M,k) in  {  (a.e-eq-class_Lp (f,M,k)) where f is PartFunc of X,REAL : f in Lp_Functions (M,k)  }  ;
hence   {  (a.e-eq-class_Lp (f,M,k)) where f is PartFunc of X,REAL : f in Lp_Functions (M,k)  }  is non empty Subset-Family of (Lp_Functions (M,k)) by A1; ::_thesis: verum
end;
end;

:: deftheorem  defines CosetSet LPSPACE2:def_7_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds CosetSet (M,k) =  {  (a.e-eq-class_Lp (f,M,k)) where f is PartFunc of X,REAL : f in Lp_Functions (M,k)  }  ;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func addCoset (M,k) ->  BinOp of (CosetSet (M,k)) means :Def8: :: LPSPACE2:def 8
 for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
it . (A,B) = a.e-eq-class_Lp ((a + b),M,k);
existence
 ex b1 being BinOp of (CosetSet (M,k)) st
for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
b1 . (A,B) = a.e-eq-class_Lp ((a + b),M,k)
proof
set C = CosetSet (M,k);
defpred S1[ set , set , set ] means  for a, b being PartFunc of X,REAL st a in $1 & b in $2 holds
$3 = a.e-eq-class_Lp ((a + b),M,k);
A1: now__::_thesis:_for_A,_B_being_Element_of_CosetSet_(M,k)_ex_z_being_Element_of_CosetSet_(M,k)_st_S1[A,B,z]
let A, B be Element of CosetSet (M,k); ::_thesis: ex z being Element of CosetSet (M,k) st S1[A,B,z]
 A in CosetSet (M,k) ;
then consider a being PartFunc of X,REAL such that
A2: ( A = a.e-eq-class_Lp (a,M,k) & a in Lp_Functions (M,k) ) ;
A3: ex E being Element of S st
( M . (E `) = 0 & dom a = E & a is_measurable_on E ) by A2, Th35;
 B in CosetSet (M,k) ;
then consider b being PartFunc of X,REAL such that
A4: ( B = a.e-eq-class_Lp (b,M,k) & b in Lp_Functions (M,k) ) ;
A5: ex E being Element of S st
( M . (E `) = 0 & dom b = E & b is_measurable_on E ) by A4, Th35;
set z = a.e-eq-class_Lp ((a + b),M,k);
 a + b in Lp_Functions (M,k) by Th25, A2, A4;
then  a.e-eq-class_Lp ((a + b),M,k) in CosetSet (M,k) ;
then reconsider z = a.e-eq-class_Lp ((a + b),M,k) as Element of CosetSet (M,k) ;
take z = z; ::_thesis: S1[A,B,z]
now__::_thesis:_for_a1,_b1_being_PartFunc_of_X,REAL_st_a1_in_A_&_b1_in_B_holds_
z_=_a.e-eq-class_Lp_((a1_+_b1),M,k)
let a1, b1 be PartFunc of X,REAL; ::_thesis: ( a1 in A & b1 in B implies z = a.e-eq-class_Lp ((a1 + b1),M,k) )
assume  ( a1 in A & b1 in B ) ; ::_thesis: z = a.e-eq-class_Lp ((a1 + b1),M,k)
then  ( a1 a.e.= a,M & b1 a.e.= b,M ) by A2, A3, A4, A5, Th37;
hence  z = a.e-eq-class_Lp ((a1 + b1),M,k) by Th42, LPSPACE1:31; ::_thesis: verum
end;
hence  S1[A,B,z] ; ::_thesis: verum
end;
consider f being Function of [:(CosetSet (M,k)),(CosetSet (M,k)):],(CosetSet (M,k)) such that
A6: for A, B being Element of CosetSet (M,k) holds S1[A,B,f . (A,B)] from BINOP_1:sch_3(A1);
reconsider f = f as BinOp of (CosetSet (M,k)) ;
take  f ; ::_thesis: for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
f . (A,B) = a.e-eq-class_Lp ((a + b),M,k)
let A, B be Element of CosetSet (M,k); ::_thesis: for a, b being PartFunc of X,REAL st a in A & b in B holds
f . (A,B) = a.e-eq-class_Lp ((a + b),M,k)
let a, b be PartFunc of X,REAL; ::_thesis: ( a in A & b in B implies f . (A,B) = a.e-eq-class_Lp ((a + b),M,k) )
assume  ( a in A & b in B ) ; ::_thesis: f . (A,B) = a.e-eq-class_Lp ((a + b),M,k)
hence  f . (A,B) = a.e-eq-class_Lp ((a + b),M,k) by A6; ::_thesis: verum
end;
uniqueness
 for b1, b2 being BinOp of (CosetSet (M,k)) st ( for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
b1 . (A,B) = a.e-eq-class_Lp ((a + b),M,k) ) & ( for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
b2 . (A,B) = a.e-eq-class_Lp ((a + b),M,k) ) holds
b1 = b2
proof
let f1, f2 be BinOp of (CosetSet (M,k)); ::_thesis: ( ( for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
f1 . (A,B) = a.e-eq-class_Lp ((a + b),M,k) ) & ( for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
f2 . (A,B) = a.e-eq-class_Lp ((a + b),M,k) ) implies f1 = f2 )
assume that
A7: for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
f1 . (A,B) = a.e-eq-class_Lp ((a + b),M,k) and
A8: for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
f2 . (A,B) = a.e-eq-class_Lp ((a + b),M,k) ; ::_thesis: f1 = f2
now__::_thesis:_for_A,_B_being_Element_of_CosetSet_(M,k)_holds_f1_._(A,B)_=_f2_._(A,B)
let A, B be Element of CosetSet (M,k); ::_thesis: f1 . (A,B) = f2 . (A,B)
 A in CosetSet (M,k) ;
then consider a1 being PartFunc of X,REAL such that
A9: ( A = a.e-eq-class_Lp (a1,M,k) & a1 in Lp_Functions (M,k) ) ;
 B in CosetSet (M,k) ;
then consider b1 being PartFunc of X,REAL such that
A10: ( B = a.e-eq-class_Lp (b1,M,k) & b1 in Lp_Functions (M,k) ) ;
A11: ( a1 in A & b1 in B ) by A9, A10, Th38;
then  f1 . (A,B) = a.e-eq-class_Lp ((a1 + b1),M,k) by A7;
hence  f1 . (A,B) = f2 . (A,B) by A8, A11; ::_thesis: verum
end;
hence  f1 = f2 by BINOP_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def8 defines addCoset LPSPACE2:def_8_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for b5 being BinOp of (CosetSet (M,k)) holds
( b5 = addCoset (M,k) iff for A, B being Element of CosetSet (M,k)
for a, b being PartFunc of X,REAL st a in A & b in B holds
b5 . (A,B) = a.e-eq-class_Lp ((a + b),M,k) );

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func zeroCoset (M,k) ->  Element of CosetSet (M,k) equals :: LPSPACE2:def 9
 a.e-eq-class_Lp ((X --> 0),M,k);
correctness
coherence
a.e-eq-class_Lp ((X --> 0),M,k) is Element of CosetSet (M,k);
proof
 X --> 0 in Lp_Functions (M,k) by Th23;
then  a.e-eq-class_Lp ((X --> 0),M,k) in CosetSet (M,k) ;
hence  a.e-eq-class_Lp ((X --> 0),M,k) is Element of CosetSet (M,k) ; ::_thesis: verum
end;
end;

:: deftheorem  defines zeroCoset LPSPACE2:def_9_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds zeroCoset (M,k) = a.e-eq-class_Lp ((X --> 0),M,k);

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func lmultCoset (M,k) ->  Function of [:REAL,(CosetSet (M,k)):],(CosetSet (M,k)) means :Def10: :: LPSPACE2:def 10
 for z being Element of REAL
for A being Element of CosetSet (M,k)
for f being PartFunc of X,REAL st f in A holds
it . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k);
existence
 ex b1 being Function of [:REAL,(CosetSet (M,k)):],(CosetSet (M,k)) st
for z being Element of REAL
for A being Element of CosetSet (M,k)
for f being PartFunc of X,REAL st f in A holds
b1 . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k)
proof
set C = CosetSet (M,k);
defpred S1[ Element of REAL , set , set ] means  for f being PartFunc of X,REAL st f in $2 holds
$3 = a.e-eq-class_Lp (($1 (#) f),M,k);
A1: now__::_thesis:_for_z_being_Element_of_REAL_
for_A_being_Element_of_CosetSet_(M,k)_ex_c_being_Element_of_CosetSet_(M,k)_st_S1[z,A,c]
let z be Element of REAL ; ::_thesis: for A being Element of CosetSet (M,k) ex c being Element of CosetSet (M,k) st S1[z,A,c]
let A be Element of CosetSet (M,k); ::_thesis: ex c being Element of CosetSet (M,k) st S1[z,A,c]
 A in CosetSet (M,k) ;
then consider a being PartFunc of X,REAL such that
A2: ( A = a.e-eq-class_Lp (a,M,k) & a in Lp_Functions (M,k) ) ;
A3: ex E being Element of S st
( M . (E `) = 0 & E = dom a & a is_measurable_on E ) by A2, Th35;
set c = a.e-eq-class_Lp ((z (#) a),M,k);
 z (#) a in Lp_Functions (M,k) by Th26, A2;
then  a.e-eq-class_Lp ((z (#) a),M,k) in CosetSet (M,k) ;
then reconsider c = a.e-eq-class_Lp ((z (#) a),M,k) as Element of CosetSet (M,k) ;
take c = c; ::_thesis: S1[z,A,c]
now__::_thesis:_for_a1_being_PartFunc_of_X,REAL_st_a1_in_A_holds_
c_=_a.e-eq-class_Lp_((z_(#)_a1),M,k)
let a1 be PartFunc of X,REAL; ::_thesis: ( a1 in A implies c = a.e-eq-class_Lp ((z (#) a1),M,k) )
assume  a1 in A ; ::_thesis: c = a.e-eq-class_Lp ((z (#) a1),M,k)
then  z (#) a1 a.e.= z (#) a,M by A2, A3, Th37, LPSPACE1:32;
hence  c = a.e-eq-class_Lp ((z (#) a1),M,k) by Th42; ::_thesis: verum
end;
hence  S1[z,A,c] ; ::_thesis: verum
end;
consider f being Function of [:REAL,(CosetSet (M,k)):],(CosetSet (M,k)) such that
A4: for z being Element of REAL
for A being Element of CosetSet (M,k) holds S1[z,A,f . (z,A)] from BINOP_1:sch_3(A1);
take  f ; ::_thesis: for z being Element of REAL
for A being Element of CosetSet (M,k)
for f being PartFunc of X,REAL st f in A holds
f . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k)
let z be Element of REAL ; ::_thesis: for A being Element of CosetSet (M,k)
for f being PartFunc of X,REAL st f in A holds
f . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k)
let A be Element of CosetSet (M,k); ::_thesis: for f being PartFunc of X,REAL st f in A holds
f . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k)
let a be PartFunc of X,REAL; ::_thesis: ( a in A implies f . (z,A) = a.e-eq-class_Lp ((z (#) a),M,k) )
assume  a in A ; ::_thesis: f . (z,A) = a.e-eq-class_Lp ((z (#) a),M,k)
hence  f . (z,A) = a.e-eq-class_Lp ((z (#) a),M,k) by A4; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of [:REAL,(CosetSet (M,k)):],(CosetSet (M,k)) st ( for z being Element of REAL
for A being Element of CosetSet (M,k)
for f being PartFunc of X,REAL st f in A holds
b1 . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k) ) & ( for z being Element of REAL
for A being Element of CosetSet (M,k)
for f being PartFunc of X,REAL st f in A holds
b2 . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k) ) holds
b1 = b2
proof
set C = CosetSet (M,k);
let f1, f2 be Function of [:REAL,(CosetSet (M,k)):],(CosetSet (M,k)); ::_thesis: ( ( for z being Element of REAL
for A being Element of CosetSet (M,k)
for f being PartFunc of X,REAL st f in A holds
f1 . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k) ) & ( for z being Element of REAL
for A being Element of CosetSet (M,k)
for f being PartFunc of X,REAL st f in A holds
f2 . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k) ) implies f1 = f2 )
assume that
A5: for z being Element of REAL
for A being Element of CosetSet (M,k)
for a being PartFunc of X,REAL st a in A holds
f1 . (z,A) = a.e-eq-class_Lp ((z (#) a),M,k) and
A6: for z being Element of REAL
for A being Element of CosetSet (M,k)
for a being PartFunc of X,REAL st a in A holds
f2 . (z,A) = a.e-eq-class_Lp ((z (#) a),M,k) ; ::_thesis: f1 = f2
now__::_thesis:_for_z_being_Element_of_REAL_
for_A_being_Element_of_CosetSet_(M,k)_holds_f1_._(z,A)_=_f2_._(z,A)
let z be Element of REAL ; ::_thesis: for A being Element of CosetSet (M,k) holds f1 . (z,A) = f2 . (z,A)
let A be Element of CosetSet (M,k); ::_thesis: f1 . (z,A) = f2 . (z,A)
 A in CosetSet (M,k) ;
then consider a1 being PartFunc of X,REAL such that
A7: ( A = a.e-eq-class_Lp (a1,M,k) & a1 in Lp_Functions (M,k) ) ;
thus f1 . (z,A) = a.e-eq-class_Lp ((z (#) a1),M,k) by A5, A7, Th38
.= f2 . (z,A) by A6, A7, Th38 ; ::_thesis: verum
end;
hence  f1 = f2 by BINOP_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def10 defines lmultCoset LPSPACE2:def_10_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for b5 being Function of [:REAL,(CosetSet (M,k)):],(CosetSet (M,k)) holds
( b5 = lmultCoset (M,k) iff for z being Element of REAL
for A being Element of CosetSet (M,k)
for f being PartFunc of X,REAL st f in A holds
b5 . (z,A) = a.e-eq-class_Lp ((z (#) f),M,k) );

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func Pre-Lp-Space (M,k) ->  strict RLSStruct  means :Def11: :: LPSPACE2:def 11
( the carrier of it = CosetSet (M,k) & the addF of it = addCoset (M,k) & 0. it = zeroCoset (M,k) & the Mult of it = lmultCoset (M,k) );
existence
 ex b1 being strict RLSStruct st
( the carrier of b1 = CosetSet (M,k) & the addF of b1 = addCoset (M,k) & 0. b1 = zeroCoset (M,k) & the Mult of b1 = lmultCoset (M,k) )
proof
take  RLSStruct(# (CosetSet (M,k)),(zeroCoset (M,k)),(addCoset (M,k)),(lmultCoset (M,k)) #) ; ::_thesis: ( the carrier of RLSStruct(# (CosetSet (M,k)),(zeroCoset (M,k)),(addCoset (M,k)),(lmultCoset (M,k)) #) = CosetSet (M,k) & the addF of RLSStruct(# (CosetSet (M,k)),(zeroCoset (M,k)),(addCoset (M,k)),(lmultCoset (M,k)) #) = addCoset (M,k) & 0. RLSStruct(# (CosetSet (M,k)),(zeroCoset (M,k)),(addCoset (M,k)),(lmultCoset (M,k)) #) = zeroCoset (M,k) & the Mult of RLSStruct(# (CosetSet (M,k)),(zeroCoset (M,k)),(addCoset (M,k)),(lmultCoset (M,k)) #) = lmultCoset (M,k) )
thus  ( the carrier of RLSStruct(# (CosetSet (M,k)),(zeroCoset (M,k)),(addCoset (M,k)),(lmultCoset (M,k)) #) = CosetSet (M,k) & the addF of RLSStruct(# (CosetSet (M,k)),(zeroCoset (M,k)),(addCoset (M,k)),(lmultCoset (M,k)) #) = addCoset (M,k) & 0. RLSStruct(# (CosetSet (M,k)),(zeroCoset (M,k)),(addCoset (M,k)),(lmultCoset (M,k)) #) = zeroCoset (M,k) & the Mult of RLSStruct(# (CosetSet (M,k)),(zeroCoset (M,k)),(addCoset (M,k)),(lmultCoset (M,k)) #) = lmultCoset (M,k) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict RLSStruct st the carrier of b1 = CosetSet (M,k) & the addF of b1 = addCoset (M,k) & 0. b1 = zeroCoset (M,k) & the Mult of b1 = lmultCoset (M,k) & the carrier of b2 = CosetSet (M,k) & the addF of b2 = addCoset (M,k) & 0. b2 = zeroCoset (M,k) & the Mult of b2 = lmultCoset (M,k) holds
b1 = b2 ;
end;

:: deftheorem Def11 defines Pre-Lp-Space LPSPACE2:def_11_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for b5 being strict RLSStruct holds
( b5 = Pre-Lp-Space (M,k) iff ( the carrier of b5 = CosetSet (M,k) & the addF of b5 = addCoset (M,k) & 0. b5 = zeroCoset (M,k) & the Mult of b5 = lmultCoset (M,k) ) );

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
cluster Pre-Lp-Space (M,k) ->  non empty strict ;
coherence
 not Pre-Lp-Space (M,k) is empty
proof
 the carrier of (Pre-Lp-Space (M,k)) = CosetSet (M,k) by Def11;
hence  not Pre-Lp-Space (M,k) is empty ; ::_thesis: verum
end;
end;

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
cluster Pre-Lp-Space (M,k) ->  right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
coherence
 ( Pre-Lp-Space (M,k) is Abelian & Pre-Lp-Space (M,k) is add-associative & Pre-Lp-Space (M,k) is right_zeroed & Pre-Lp-Space (M,k) is right_complementable & Pre-Lp-Space (M,k) is vector-distributive & Pre-Lp-Space (M,k) is scalar-distributive & Pre-Lp-Space (M,k) is scalar-associative & Pre-Lp-Space (M,k) is scalar-unital )
proof
set C = CosetSet (M,k);
set aC = addCoset (M,k);
set lC = lmultCoset (M,k);
set A = Pre-Lp-Space (M,k);
A1: ( the carrier of (Pre-Lp-Space (M,k)) = CosetSet (M,k) & the addF of (Pre-Lp-Space (M,k)) = addCoset (M,k) & 0. (Pre-Lp-Space (M,k)) = zeroCoset (M,k) & the Mult of (Pre-Lp-Space (M,k)) = lmultCoset (M,k) ) by Def11;
thus  Pre-Lp-Space (M,k) is Abelian ::_thesis: ( Pre-Lp-Space (M,k) is add-associative & Pre-Lp-Space (M,k) is right_zeroed & Pre-Lp-Space (M,k) is right_complementable & Pre-Lp-Space (M,k) is vector-distributive & Pre-Lp-Space (M,k) is scalar-distributive & Pre-Lp-Space (M,k) is scalar-associative & Pre-Lp-Space (M,k) is scalar-unital )
proof
let A1, A2 be Element of (Pre-Lp-Space (M,k)); :: according to RLVECT_1:def_2 ::_thesis: A1 + A2 = A2 + A1
 A1 in CosetSet (M,k) by A1;
then consider a being PartFunc of X,REAL such that
A2: ( A1 = a.e-eq-class_Lp (a,M,k) & a in Lp_Functions (M,k) ) ;
 A2 in CosetSet (M,k) by A1;
then consider b being PartFunc of X,REAL such that
A3: ( A2 = a.e-eq-class_Lp (b,M,k) & b in Lp_Functions (M,k) ) ;
A4: ( a in A1 & b in A2 ) by A2, A3, Th38;
then  A1 + A2 = a.e-eq-class_Lp ((a + b),M,k) by A1, Def8;
hence  A1 + A2 = A2 + A1 by A1, A4, Def8; ::_thesis: verum
end;
thus  Pre-Lp-Space (M,k) is add-associative ::_thesis: ( Pre-Lp-Space (M,k) is right_zeroed & Pre-Lp-Space (M,k) is right_complementable & Pre-Lp-Space (M,k) is vector-distributive & Pre-Lp-Space (M,k) is scalar-distributive & Pre-Lp-Space (M,k) is scalar-associative & Pre-Lp-Space (M,k) is scalar-unital )
proof
let A1, A2, A3 be Element of (Pre-Lp-Space (M,k)); :: according to RLVECT_1:def_3 ::_thesis: (A1 + A2) + A3 = A1 + (A2 + A3)
 A1 in CosetSet (M,k) by A1;
then consider a being PartFunc of X,REAL such that
A5: ( A1 = a.e-eq-class_Lp (a,M,k) & a in Lp_Functions (M,k) ) ;
 A2 in CosetSet (M,k) by A1;
then consider b being PartFunc of X,REAL such that
A6: ( A2 = a.e-eq-class_Lp (b,M,k) & b in Lp_Functions (M,k) ) ;
 A3 in CosetSet (M,k) by A1;
then consider c being PartFunc of X,REAL such that
A7: ( A3 = a.e-eq-class_Lp (c,M,k) & c in Lp_Functions (M,k) ) ;
A8: ( a in A1 & b in A2 & c in A3 ) by A5, A6, A7, Th38;
then  ( (addCoset (M,k)) . (A1,A2) = a.e-eq-class_Lp ((a + b),M,k) & (addCoset (M,k)) . (A2,A3) = a.e-eq-class_Lp ((b + c),M,k) ) by A1, Def8;
then A9: ( a + b in A1 + A2 & b + c in A2 + A3 ) by A1, Th38, Th25, A5, A6, A7;
reconsider a1 = a, b1 = b, c1 = c as VECTOR of (RLSp_LpFunct (M,k)) by A5, A6, A7;
A10: ( a + b = a1 + b1 & b + c = b1 + c1 ) by Th29;
then  a + (b + c) = a1 + (b1 + c1) by Th29;
then  a + (b + c) = (a1 + b1) + c1 by RLVECT_1:def_3;
then  a + (b + c) = (a + b) + c by A10, Th29;
then  (A1 + A2) + A3 = a.e-eq-class_Lp ((a + (b + c)),M,k) by A8, A9, Def8, A1;
hence  (A1 + A2) + A3 = A1 + (A2 + A3) by A8, A9, Def8, A1; ::_thesis: verum
end;
thus  Pre-Lp-Space (M,k) is right_zeroed ::_thesis: ( Pre-Lp-Space (M,k) is right_complementable & Pre-Lp-Space (M,k) is vector-distributive & Pre-Lp-Space (M,k) is scalar-distributive & Pre-Lp-Space (M,k) is scalar-associative & Pre-Lp-Space (M,k) is scalar-unital )
proof
let A1 be Element of (Pre-Lp-Space (M,k)); :: according to RLVECT_1:def_4 ::_thesis: A1 + (0. (Pre-Lp-Space (M,k))) = A1
 A1 in CosetSet (M,k) by A1;
then consider a being PartFunc of X,REAL such that
A11: ( A1 = a.e-eq-class_Lp (a,M,k) & a in Lp_Functions (M,k) ) ;
A12: a in A1 by A11, Th38;
set z = X --> 0;
A13: X --> 0 in 0. (Pre-Lp-Space (M,k)) by A1, Th38, Th23;
reconsider a1 = a, z1 = X --> 0 as VECTOR of (RLSp_LpFunct (M,k)) by A11, Th23;
a + (X --> 0) = a1 + z1 by Th29
.= a1 + (0. (RLSp_LpFunct (M,k))) by Th32
.= a by RLVECT_1:def_4 ;
hence  A1 + (0. (Pre-Lp-Space (M,k))) = A1 by A1, A11, A12, A13, Def8; ::_thesis: verum
end;
thus  Pre-Lp-Space (M,k) is right_complementable ::_thesis: ( Pre-Lp-Space (M,k) is vector-distributive & Pre-Lp-Space (M,k) is scalar-distributive & Pre-Lp-Space (M,k) is scalar-associative & Pre-Lp-Space (M,k) is scalar-unital )
proof
let A1 be Element of (Pre-Lp-Space (M,k)); :: according to ALGSTR_0:def_16 ::_thesis: A1 is right_complementable
 A1 in CosetSet (M,k) by A1;
then consider a being PartFunc of X,REAL such that
A14: ( A1 = a.e-eq-class_Lp (a,M,k) & a in Lp_Functions (M,k) ) ;
A15: a in A1 by A14, Th38;
reconsider a1 = a as VECTOR of (RLSp_LpFunct (M,k)) by A14;
A16: (- 1) (#) a in Lp_Functions (M,k) by A14, Th26;
set A2 = a.e-eq-class_Lp (((- 1) (#) a),M,k);
 a.e-eq-class_Lp (((- 1) (#) a),M,k) in CosetSet (M,k) by A16;
then reconsider A2 = a.e-eq-class_Lp (((- 1) (#) a),M,k) as Element of (Pre-Lp-Space (M,k)) by A1;
take  A2 ; :: according to ALGSTR_0:def_11 ::_thesis: A1 + A2 = 0. (Pre-Lp-Space (M,k))
A17: (- 1) (#) a in A2 by Th38, A14, Th26;
consider v, g being PartFunc of X,REAL such that
A18: ( v in Lp_Functions (M,k) & g in Lp_Functions (M,k) & v = a1 + ((- 1) * a1) & g = X --> 0 & v a.e.= g,M ) by Th31;
 (- 1) (#) a = (- 1) * a1 by Th30;
then  a + ((- 1) (#) a) a.e.= g,M by Th29, A18;
then  0. (Pre-Lp-Space (M,k)) = a.e-eq-class_Lp ((a + ((- 1) (#) a)),M,k) by Th42, A18, A1;
hence  A1 + A2 = 0. (Pre-Lp-Space (M,k)) by A15, A17, Def8, A1; ::_thesis: verum
end;
now__::_thesis:_for_x0,_y0_being_real_number_
for_A1,_A2_being_Element_of_(Pre-Lp-Space_(M,k))_holds_
(_x0_*_(A1_+_A2)_=_(x0_*_A1)_+_(x0_*_A2)_&_(x0_+_y0)_*_A1_=_(x0_*_A1)_+_(y0_*_A1)_&_(x0_*_y0)_*_A1_=_x0_*_(y0_*_A1)_&_1_*_A1_=_A1_)
let x0, y0 be real number ; ::_thesis: for A1, A2 being Element of (Pre-Lp-Space (M,k)) holds
( x0 * (A1 + A2) = (x0 * A1) + (x0 * A2) & (x0 + y0) * A1 = (x0 * A1) + (y0 * A1) & (x0 * y0) * A1 = x0 * (y0 * A1) & 1 * A1 = A1 )
let A1, A2 be Element of (Pre-Lp-Space (M,k)); ::_thesis: ( x0 * (A1 + A2) = (x0 * A1) + (x0 * A2) & (x0 + y0) * A1 = (x0 * A1) + (y0 * A1) & (x0 * y0) * A1 = x0 * (y0 * A1) & 1 * A1 = A1 )
reconsider x = x0, y = y0 as Real by XREAL_0:def_1;
 A1 in CosetSet (M,k) by A1;
then consider a being PartFunc of X,REAL such that
A19: ( A1 = a.e-eq-class_Lp (a,M,k) & a in Lp_Functions (M,k) ) ;
 A2 in CosetSet (M,k) by A1;
then consider b being PartFunc of X,REAL such that
A20: ( A2 = a.e-eq-class_Lp (b,M,k) & b in Lp_Functions (M,k) ) ;
A21: ( a in A1 & b in A2 ) by A19, A20, Th38;
then  (addCoset (M,k)) . (A1,A2) = a.e-eq-class_Lp ((a + b),M,k) by A1, Def8;
then A22: a + b in A1 + A2 by Th38, Th25, A19, A20, A1;
reconsider a1 = a, b1 = b as VECTOR of (RLSp_LpFunct (M,k)) by A19, A20;
A23: ( y (#) a = y * a1 & x (#) a = x * a1 & x (#) b = x * b1 & (x + y) (#) a = (x + y) * a1 & 1 (#) a = 1 * a1 ) by Th30;
 a + b = a1 + b1 by Th29;
then  x (#) (a + b) = x * (a1 + b1) by Th30;
then  x (#) (a + b) = (x * a1) + (x * b1) by RLVECT_1:def_5;
then A24: x (#) (a + b) = (x (#) a) + (x (#) b) by A23, Th29;
 (x + y) (#) a = (x * a1) + (y * a1) by A23, RLVECT_1:def_6;
then A25: (x + y) (#) a = (x (#) a) + (y (#) a) by A23, Th29;
x (#) (y (#) a) = x * (y * a1) by A23, Th30
.= (x * y) * a1 by RLVECT_1:def_7 ;
then A26: x (#) (y (#) a) = (x * y) (#) a by Th30;
 ( (lmultCoset (M,k)) . (x,A1) = a.e-eq-class_Lp ((x (#) a),M,k) & (lmultCoset (M,k)) . (x,A2) = a.e-eq-class_Lp ((x (#) b),M,k) & (lmultCoset (M,k)) . (y,A1) = a.e-eq-class_Lp ((y (#) a),M,k) ) by A1, A21, Def10;
then A27: ( x (#) a in x * A1 & x (#) b in x * A2 & y (#) a in y * A1 ) by A1, Th38, Th26, A19, A20;
 x * (A1 + A2) = a.e-eq-class_Lp (((x (#) a) + (x (#) b)),M,k) by A1, A24, A22, Def10;
hence  x0 * (A1 + A2) = (x0 * A1) + (x0 * A2) by A1, A27, Def8; ::_thesis: ( (x0 + y0) * A1 = (x0 * A1) + (y0 * A1) & (x0 * y0) * A1 = x0 * (y0 * A1) & 1 * A1 = A1 )
 (x + y) * A1 = a.e-eq-class_Lp (((x (#) a) + (y (#) a)),M,k) by A1, A25, A21, Def10;
hence  (x0 + y0) * A1 = (x0 * A1) + (y0 * A1) by A27, Def8, A1; ::_thesis: ( (x0 * y0) * A1 = x0 * (y0 * A1) & 1 * A1 = A1 )
 (x0 * y0) * A1 = a.e-eq-class_Lp ((x (#) (y (#) a)),M,k) by A1, A26, A21, Def10;
hence  (x0 * y0) * A1 = x0 * (y0 * A1) by A27, Def10, A1; ::_thesis: 1 * A1 = A1
 1 (#) a = a by A23, RLVECT_1:def_8;
hence  1 * A1 = A1 by A19, A21, Def10, A1; ::_thesis: verum
end;
hence  ( Pre-Lp-Space (M,k) is vector-distributive & Pre-Lp-Space (M,k) is scalar-distributive & Pre-Lp-Space (M,k) is scalar-associative & Pre-Lp-Space (M,k) is scalar-unital ) by RLVECT_1:def_5, RLVECT_1:def_6, RLVECT_1:def_7, RLVECT_1:def_8; ::_thesis: verum
end;
end;

begin

theorem Th48: :: LPSPACE2:48
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & f a.e.= g,M holds
Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k))
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & f a.e.= g,M holds
Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k))
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & f a.e.= g,M holds
Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k))
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & f a.e.= g,M holds
Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k))
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & f a.e.= g,M holds
Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k))
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & f a.e.= g,M implies Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
set t = (abs f) to_power k;
set s = (abs g) to_power k;
assume A1: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & f a.e.= g,M ) ; ::_thesis: Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k))
then  ex f1 being PartFunc of X,REAL st
( f = f1 & ex E being Element of S st
( M . (E `) = 0 & dom f1 = E & f1 is_measurable_on E & (abs f1) to_power k is_integrable_on M ) ) ;
then consider Df being Element of S such that
A2: ( M . (Df `) = 0 & dom f = Df & f is_measurable_on Df & (abs f) to_power k is_integrable_on M ) ;
 ex g1 being PartFunc of X,REAL st
( g = g1 & ex E being Element of S st
( M . (E `) = 0 & dom g1 = E & g1 is_measurable_on E & (abs g1) to_power k is_integrable_on M ) ) by A1;
then consider Dg being Element of S such that
A3: ( M . (Dg `) = 0 & dom g = Dg & g is_measurable_on Dg & (abs g) to_power k is_integrable_on M ) ;
A4: ( dom (abs f) = dom f & dom (abs g) = dom g ) by VALUED_1:def_11;
consider E1 being Element of S such that
A5: ( M . E1 = 0 & f | (E1 `) = g | (E1 `) ) by A1, LPSPACE1:def_10;
reconsider NDf = Df ` , NDg = Dg ` as Element of S by MEASURE1:34;
set Ef = Df \ (NDg \/ E1);
set Eg = Dg \ (NDf \/ E1);
set E2 = (NDf \/ NDg) \/ E1;
 ( NDf is measure_zero of M & NDg is measure_zero of M & E1 is measure_zero of M ) by A2, A3, A5, MEASURE1:def_7;
then  ( NDf \/ E1 is measure_zero of M & NDg \/ E1 is measure_zero of M ) by MEASURE1:37;
then A6: ( M . (NDf \/ E1) = 0 & M . (NDg \/ E1) = 0 ) by MEASURE1:def_7;
 ( X \ NDf = X /\ Df & X \ NDg = X /\ Dg ) by XBOOLE_1:48;
then A7: ( X \ NDf = Df & X \ NDg = Dg ) by XBOOLE_1:28;
 ( Df \ (NDg \/ E1) = (Df \ NDg) \ E1 & Dg \ (NDf \/ E1) = (Dg \ NDf) \ E1 ) by XBOOLE_1:41;
then A8: ( Df \ (NDg \/ E1) = (X \ (NDf \/ NDg)) \ E1 & Dg \ (NDf \/ E1) = (X \ (NDf \/ NDg)) \ E1 ) by A7, XBOOLE_1:41;
then A9: ( Df \ (NDg \/ E1) = X \ ((NDf \/ NDg) \/ E1) & Dg \ (NDf \/ E1) = X \ ((NDf \/ NDg) \/ E1) ) by XBOOLE_1:41;
 ( abs f is_measurable_on Df & abs g is_measurable_on Dg ) by A2, A3, MESFUNC6:48;
then A10: ( (abs f) to_power k is_measurable_on Df & (abs g) to_power k is_measurable_on Dg ) by A2, A3, A4, MESFUN6C:29;
A11: ( dom ((abs f) to_power k) = Df & dom ((abs g) to_power k) = Dg ) by A2, A3, A4, MESFUN6C:def_4;
then A12: ( Integral (M,(((abs f) to_power k) | (Df \ (NDg \/ E1)))) = Integral (M,((abs f) to_power k)) & Integral (M,(((abs g) to_power k) | (Dg \ (NDf \/ E1)))) = Integral (M,((abs g) to_power k)) ) by A6, A10, MESFUNC6:89;
 ( dom (((abs f) to_power k) | (Df \ (NDg \/ E1))) = (dom ((abs f) to_power k)) /\ (Df \ (NDg \/ E1)) & dom (((abs g) to_power k) | (Df \ (NDg \/ E1))) = (dom ((abs g) to_power k)) /\ (Df \ (NDg \/ E1)) ) by RELAT_1:61;
then A13: ( dom (((abs f) to_power k) | (Df \ (NDg \/ E1))) = (Df /\ Df) \ (NDg \/ E1) & dom (((abs g) to_power k) | (Df \ (NDg \/ E1))) = (Dg /\ Dg) \ (NDf \/ E1) ) by A11, A8, XBOOLE_1:49;
now__::_thesis:_for_x_being_Element_of_X_st_x_in_dom_(((abs_f)_to_power_k)_|_(Df_\_(NDg_\/_E1)))_holds_
(((abs_f)_to_power_k)_|_(Df_\_(NDg_\/_E1)))_._x_=_(((abs_g)_to_power_k)_|_(Df_\_(NDg_\/_E1)))_._x
let x be Element of X; ::_thesis: ( x in dom (((abs f) to_power k) | (Df \ (NDg \/ E1))) implies (((abs f) to_power k) | (Df \ (NDg \/ E1))) . x = (((abs g) to_power k) | (Df \ (NDg \/ E1))) . x )
assume A14: x in dom (((abs f) to_power k) | (Df \ (NDg \/ E1))) ; ::_thesis: (((abs f) to_power k) | (Df \ (NDg \/ E1))) . x = (((abs g) to_power k) | (Df \ (NDg \/ E1))) . x
A15: ( dom (((abs f) to_power k) | (Df \ (NDg \/ E1))) c= dom ((abs f) to_power k) & dom (((abs g) to_power k) | (Df \ (NDg \/ E1))) c= dom ((abs g) to_power k) ) by RELAT_1:60;
 ((NDf \/ NDg) \/ E1) ` c= E1 ` by XBOOLE_1:7, XBOOLE_1:34;
then A16: ( f . x = (f | (E1 `)) . x & g . x = (g | (E1 `)) . x ) by A14, A13, A9, FUNCT_1:49;
 ( (((abs f) to_power k) | (Df \ (NDg \/ E1))) . x = ((abs f) to_power k) . x & (((abs g) to_power k) | (Df \ (NDg \/ E1))) . x = ((abs g) to_power k) . x ) by A14, A13, FUNCT_1:49;
then  ( (((abs f) to_power k) | (Df \ (NDg \/ E1))) . x = ((abs f) . x) to_power k & (((abs g) to_power k) | (Df \ (NDg \/ E1))) . x = ((abs g) . x) to_power k ) by A8, A13, A14, A15, MESFUN6C:def_4;
then  ( (((abs f) to_power k) | (Df \ (NDg \/ E1))) . x = (abs (f . x)) to_power k & (((abs g) to_power k) | (Df \ (NDg \/ E1))) . x = (abs (g . x)) to_power k ) by VALUED_1:18;
hence  (((abs f) to_power k) | (Df \ (NDg \/ E1))) . x = (((abs g) to_power k) | (Df \ (NDg \/ E1))) . x by A5, A16; ::_thesis: verum
end;
hence  Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) by A12, A13, A8, PARTFUN1:5; ::_thesis: verum
end;

theorem Th49: :: LPSPACE2:49
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) )
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) )
let f be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) holds
( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) )
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) implies ( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) ) )
assume  f in Lp_Functions (M,k) ; ::_thesis: ( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) )
then A1: ex f1 being PartFunc of X,REAL st
( f = f1 & ex ND being Element of S st
( M . (ND `) = 0 & dom f1 = ND & f1 is_measurable_on ND & (abs f1) to_power k is_integrable_on M ) ) ;
then  ( -infty < Integral (M,((abs f) to_power k)) & Integral (M,((abs f) to_power k)) < +infty ) by MESFUNC6:90;
hence  Integral (M,((abs f) to_power k)) in REAL by XXREAL_0:14; ::_thesis: 0 <= Integral (M,((abs f) to_power k))
 R_EAL ((abs f) to_power k) is_integrable_on M by A1, MESFUNC6:def_4;
then consider A being Element of S such that
A2: ( A = dom (R_EAL ((abs f) to_power k)) & R_EAL ((abs f) to_power k) is_measurable_on A ) by MESFUNC5:def_17;
 ( A = dom ((abs f) to_power k) & (abs f) to_power k is_measurable_on A ) by A2, MESFUNC6:def_1;
hence  0 <= Integral (M,((abs f) to_power k)) by MESFUNC6:84; ::_thesis: verum
end;

theorem Th50: :: LPSPACE2:50
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st ex x being VECTOR of (Pre-Lp-Space (M,k)) st
( f in x & g in x ) holds
( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st ex x being VECTOR of (Pre-Lp-Space (M,k)) st
( f in x & g in x ) holds
( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st ex x being VECTOR of (Pre-Lp-Space (M,k)) st
( f in x & g in x ) holds
( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) )
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real st ex x being VECTOR of (Pre-Lp-Space (M,k)) st
( f in x & g in x ) holds
( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) )
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real st ex x being VECTOR of (Pre-Lp-Space (M,k)) st
( f in x & g in x ) holds
( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) )
let k be positive Real; ::_thesis: ( ex x being VECTOR of (Pre-Lp-Space (M,k)) st
( f in x & g in x ) implies ( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) )
assume  ex x being VECTOR of (Pre-Lp-Space (M,k)) st
( f in x & g in x ) ; ::_thesis: ( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) )
then consider x being VECTOR of (Pre-Lp-Space (M,k)) such that
A1: ( f in x & g in x ) ;
 x in the carrier of (Pre-Lp-Space (M,k)) ;
then  x in CosetSet (M,k) by Def11;
then consider h being PartFunc of X,REAL such that
A2: ( x = a.e-eq-class_Lp (h,M,k) & h in Lp_Functions (M,k) ) ;
 ( ex i being PartFunc of X,REAL st
( f = i & i in Lp_Functions (M,k) & h a.e.= i,M ) & ex j being PartFunc of X,REAL st
( g = j & j in Lp_Functions (M,k) & h a.e.= j,M ) ) by A1, A2;
then  ( f a.e.= h,M & h a.e.= g,M ) by LPSPACE1:29;
hence  ( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) by A1, A2, LPSPACE1:30; ::_thesis: verum
end;

theorem Th51: :: LPSPACE2:51
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
let f be PartFunc of X,REAL; ::_thesis: for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
let k be positive Real; ::_thesis: for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
let x be Point of (Pre-Lp-Space (M,k)); ::_thesis: ( f in x implies ( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) ) )
assume A1: f in x ; ::_thesis: ( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
 x in the carrier of (Pre-Lp-Space (M,k)) ;
then  x in CosetSet (M,k) by Def11;
then consider h being PartFunc of X,REAL such that
A2: ( x = a.e-eq-class_Lp (h,M,k) & h in Lp_Functions (M,k) ) ;
 ex g being PartFunc of X,REAL st
( f = g & g in Lp_Functions (M,k) & h a.e.= g,M ) by A1, A2;
then  ex f0 being PartFunc of X,REAL st
( f = f0 & ex ND being Element of S st
( M . (ND `) = 0 & dom f0 = ND & f0 is_measurable_on ND & (abs f0) to_power k is_integrable_on M ) ) ;
hence  ( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) ) ; ::_thesis: verum
end;

theorem Th52: :: LPSPACE2:52
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let f, g be PartFunc of X,REAL; ::_thesis: for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let k be positive Real; ::_thesis: for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let x be Point of (Pre-Lp-Space (M,k)); ::_thesis: ( f in x & g in x implies ( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) ) )
assume  ( f in x & g in x ) ; ::_thesis: ( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
then  ( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) by Th50;
hence  ( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) ) by Th48; ::_thesis: verum
end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func Lp-Norm (M,k) ->  Function of the carrier of (Pre-Lp-Space (M,k)),REAL means :Def12: :: LPSPACE2:def 12
 for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & it . x = r to_power (1 / k) ) );
existence
 ex b1 being Function of the carrier of (Pre-Lp-Space (M,k)),REAL st
for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & b1 . x = r to_power (1 / k) ) )
proof
defpred S1[ set , set ] means  ex f being PartFunc of X,REAL st
( f in $1 & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & $2 = r to_power (1 / k) ) );
A1: for x being Point of (Pre-Lp-Space (M,k)) ex y being Element of REAL st S1[x,y]
proof
let x be Point of (Pre-Lp-Space (M,k)); ::_thesis: ex y being Element of REAL st S1[x,y]
 x in the carrier of (Pre-Lp-Space (M,k)) ;
then  x in CosetSet (M,k) by Def11;
then consider f being PartFunc of X,REAL such that
A2: ( x = a.e-eq-class_Lp (f,M,k) & f in Lp_Functions (M,k) ) ;
reconsider r1 = Integral (M,((abs f) to_power k)) as Element of REAL by A2, Th49;
 r1 to_power (1 / k) in REAL ;
hence  ex y being Element of REAL st S1[x,y] by A2, Th38; ::_thesis: verum
end;
consider F being Function of the carrier of (Pre-Lp-Space (M,k)),REAL such that
A3: for x being Point of (Pre-Lp-Space (M,k)) holds S1[x,F . x] from FUNCT_2:sch_3(A1);
take  F ; ::_thesis: for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & F . x = r to_power (1 / k) ) )
thus  for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & F . x = r to_power (1 / k) ) ) by A3; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of the carrier of (Pre-Lp-Space (M,k)),REAL st ( for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & b1 . x = r to_power (1 / k) ) ) ) & ( for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & b2 . x = r to_power (1 / k) ) ) ) holds
b1 = b2
proof
let N1, N2 be Function of the carrier of (Pre-Lp-Space (M,k)),REAL; ::_thesis: ( ( for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & N1 . x = r to_power (1 / k) ) ) ) & ( for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & N2 . x = r to_power (1 / k) ) ) ) implies N1 = N2 )
assume A4: ( ( for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r1 being Real st
( r1 = Integral (M,((abs f) to_power k)) & N1 . x = r1 to_power (1 / k) ) ) ) & ( for x being Point of (Pre-Lp-Space (M,k)) ex g being PartFunc of X,REAL st
( g in x & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power k)) & N2 . x = r2 to_power (1 / k) ) ) ) ) ; ::_thesis: N1 = N2
now__::_thesis:_for_x_being_Point_of_(Pre-Lp-Space_(M,k))_holds_N1_._x_=_N2_._x
let x be Point of (Pre-Lp-Space (M,k)); ::_thesis: N1 . x = N2 . x
 ( ex f being PartFunc of X,REAL st
( f in x & ex r1 being Real st
( r1 = Integral (M,((abs f) to_power k)) & N1 . x = r1 to_power (1 / k) ) ) & ex g being PartFunc of X,REAL st
( g in x & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power k)) & N2 . x = r2 to_power (1 / k) ) ) ) by A4;
hence  N1 . x = N2 . x by Th52; ::_thesis: verum
end;
hence  N1 = N2 by FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def12 defines Lp-Norm LPSPACE2:def_12_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for b5 being Function of the carrier of (Pre-Lp-Space (M,k)),REAL holds
( b5 = Lp-Norm (M,k) iff for x being Point of (Pre-Lp-Space (M,k)) ex f being PartFunc of X,REAL st
( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & b5 . x = r to_power (1 / k) ) ) );

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be positive Real;
func Lp-Space (M,k) ->  non empty NORMSTR  equals :: LPSPACE2:def 13
 NORMSTR(# the carrier of (Pre-Lp-Space (M,k)), the ZeroF of (Pre-Lp-Space (M,k)), the addF of (Pre-Lp-Space (M,k)), the Mult of (Pre-Lp-Space (M,k)),(Lp-Norm (M,k)) #);
coherence
 NORMSTR(# the carrier of (Pre-Lp-Space (M,k)), the ZeroF of (Pre-Lp-Space (M,k)), the addF of (Pre-Lp-Space (M,k)), the Mult of (Pre-Lp-Space (M,k)),(Lp-Norm (M,k)) #) is non empty NORMSTR ;
end;

:: deftheorem  defines Lp-Space LPSPACE2:def_13_:_
 for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds Lp-Space (M,k) = NORMSTR(# the carrier of (Pre-Lp-Space (M,k)), the ZeroF of (Pre-Lp-Space (M,k)), the addF of (Pre-Lp-Space (M,k)), the Mult of (Pre-Lp-Space (M,k)),(Lp-Norm (M,k)) #);


theorem Th53: :: LPSPACE2:53
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for x being Point of (Lp-Space (M,k)) holds
( ex f being PartFunc of X,REAL st
( f in Lp_Functions (M,k) & x = a.e-eq-class_Lp (f,M,k) ) & ( for f being PartFunc of X,REAL st f in x holds
ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) ) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for x being Point of (Lp-Space (M,k)) holds
( ex f being PartFunc of X,REAL st
( f in Lp_Functions (M,k) & x = a.e-eq-class_Lp (f,M,k) ) & ( for f being PartFunc of X,REAL st f in x holds
ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) ) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real
for x being Point of (Lp-Space (M,k)) holds
( ex f being PartFunc of X,REAL st
( f in Lp_Functions (M,k) & x = a.e-eq-class_Lp (f,M,k) ) & ( for f being PartFunc of X,REAL st f in x holds
ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) ) )
let M be sigma_Measure of S; ::_thesis: for k being positive Real
for x being Point of (Lp-Space (M,k)) holds
( ex f being PartFunc of X,REAL st
( f in Lp_Functions (M,k) & x = a.e-eq-class_Lp (f,M,k) ) & ( for f being PartFunc of X,REAL st f in x holds
ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) ) )
let k be positive Real; ::_thesis: for x being Point of (Lp-Space (M,k)) holds
( ex f being PartFunc of X,REAL st
( f in Lp_Functions (M,k) & x = a.e-eq-class_Lp (f,M,k) ) & ( for f being PartFunc of X,REAL st f in x holds
ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) ) )
let x be Point of (Lp-Space (M,k)); ::_thesis: ( ex f being PartFunc of X,REAL st
( f in Lp_Functions (M,k) & x = a.e-eq-class_Lp (f,M,k) ) & ( for f being PartFunc of X,REAL st f in x holds
ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) ) )
 x in the carrier of (Pre-Lp-Space (M,k)) ;
then  x in CosetSet (M,k) by Def11;
then  ex g being PartFunc of X,REAL st
( x = a.e-eq-class_Lp (g,M,k) & g in Lp_Functions (M,k) ) ;
hence  ex f being PartFunc of X,REAL st
( f in Lp_Functions (M,k) & x = a.e-eq-class_Lp (f,M,k) ) ; ::_thesis: for f being PartFunc of X,REAL st f in x holds
ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) )
consider f being PartFunc of X,REAL such that
A1: ( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & (Lp-Norm (M,k)) . x = r to_power (1 / k) ) ) by Def12;
hereby  ::_thesis: verum
let g be PartFunc of X,REAL; ::_thesis: ( g in x implies ex r being Real st
( 0 <= r & r = Integral (M,((abs g) to_power k)) & ||.x.|| = r to_power (1 / k) ) )
assume A2: g in x ; ::_thesis: ex r being Real st
( 0 <= r & r = Integral (M,((abs g) to_power k)) & ||.x.|| = r to_power (1 / k) )
then A3: g in Lp_Functions (M,k) by Th50;
 Integral (M,((abs g) to_power k)) = Integral (M,((abs f) to_power k)) by A1, Th52, A2;
hence  ex r being Real st
( 0 <= r & r = Integral (M,((abs g) to_power k)) & ||.x.|| = r to_power (1 / k) ) by A1, A3, Th49; ::_thesis: verum
end;
end;

theorem Th54: :: LPSPACE2:54
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real
for x, y being Point of (Lp-Space (M,k)) holds
( ( f in x & g in y implies f + g in x + y ) & ( f in x implies a (#) f in a * x ) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real
for x, y being Point of (Lp-Space (M,k)) holds
( ( f in x & g in y implies f + g in x + y ) & ( f in x implies a (#) f in a * x ) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real
for x, y being Point of (Lp-Space (M,k)) holds
( ( f in x & g in y implies f + g in x + y ) & ( f in x implies a (#) f in a * x ) )
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real
for x, y being Point of (Lp-Space (M,k)) holds
( ( f in x & g in y implies f + g in x + y ) & ( f in x implies a (#) f in a * x ) )
let f, g be PartFunc of X,REAL; ::_thesis: for a being Real
for k being positive Real
for x, y being Point of (Lp-Space (M,k)) holds
( ( f in x & g in y implies f + g in x + y ) & ( f in x implies a (#) f in a * x ) )
let a be Real; ::_thesis: for k being positive Real
for x, y being Point of (Lp-Space (M,k)) holds
( ( f in x & g in y implies f + g in x + y ) & ( f in x implies a (#) f in a * x ) )
let k be positive Real; ::_thesis: for x, y being Point of (Lp-Space (M,k)) holds
( ( f in x & g in y implies f + g in x + y ) & ( f in x implies a (#) f in a * x ) )
let x, y be Point of (Lp-Space (M,k)); ::_thesis: ( ( f in x & g in y implies f + g in x + y ) & ( f in x implies a (#) f in a * x ) )
set C = CosetSet (M,k);
hereby  ::_thesis: ( f in x implies a (#) f in a * x )
assume A1: ( f in x & g in y ) ; ::_thesis: f + g in x + y
 x in the carrier of (Pre-Lp-Space (M,k)) ;
then A2: x in CosetSet (M,k) by Def11;
then consider a being PartFunc of X,REAL such that
A3: ( x = a.e-eq-class_Lp (a,M,k) & a in Lp_Functions (M,k) ) ;
A4: a in x by A3, Th38;
 y in the carrier of (Pre-Lp-Space (M,k)) ;
then A5: y in CosetSet (M,k) by Def11;
then consider b being PartFunc of X,REAL such that
A6: ( y = a.e-eq-class_Lp (b,M,k) & b in Lp_Functions (M,k) ) ;
 b in y by A6, Th38;
then  (addCoset (M,k)) . (x,y) = a.e-eq-class_Lp ((a + b),M,k) by A2, A5, A4, Def8;
then A7: x + y = a.e-eq-class_Lp ((a + b),M,k) by Def11;
 ex r being PartFunc of X,REAL st
( f = r & r in Lp_Functions (M,k) & a a.e.= r,M ) by A1, A3;
then A8: a.e-eq-class_Lp (a,M,k) = a.e-eq-class_Lp (f,M,k) by Th42;
 ex r being PartFunc of X,REAL st
( g = r & r in Lp_Functions (M,k) & b a.e.= r,M ) by A1, A6;
then  a.e-eq-class_Lp (b,M,k) = a.e-eq-class_Lp (g,M,k) by Th42;
then  a.e-eq-class_Lp ((a + b),M,k) = a.e-eq-class_Lp ((f + g),M,k) by A1, A3, A6, A8, Th45;
hence  f + g in x + y by Th38, A7, Th25, A3, A1, A6; ::_thesis: verum
end;
hereby  ::_thesis: verum
assume A9: f in x ; ::_thesis: a (#) f in a * x
 x in the carrier of (Pre-Lp-Space (M,k)) ;
then A10: x in CosetSet (M,k) by Def11;
then consider f1 being PartFunc of X,REAL such that
A11: ( x = a.e-eq-class_Lp (f1,M,k) & f1 in Lp_Functions (M,k) ) ;
 f1 in x by A11, Th38;
then  (lmultCoset (M,k)) . (a,x) = a.e-eq-class_Lp ((a (#) f1),M,k) by A10, Def10;
then A12: a * x = a.e-eq-class_Lp ((a (#) f1),M,k) by Def11;
 ex r being PartFunc of X,REAL st
( f = r & r in Lp_Functions (M,k) & f1 a.e.= r,M ) by A9, A11;
then  a.e-eq-class_Lp (f1,M,k) = a.e-eq-class_Lp (f,M,k) by Th42;
then  a.e-eq-class_Lp ((a (#) f1),M,k) = a.e-eq-class_Lp ((a (#) f),M,k) by A11, A9, Th47;
hence  a (#) f in a * x by A12, Th26, A9, A11, Th38; ::_thesis: verum
end;
end;

theorem Th55: :: LPSPACE2:55
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Lp-Space (M,k)) st f in x holds
( x = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Lp-Space (M,k)) st f in x holds
( x = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Lp-Space (M,k)) st f in x holds
( x = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) )
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Lp-Space (M,k)) st f in x holds
( x = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) )
let f be PartFunc of X,REAL; ::_thesis: for k being positive Real
for x being Point of (Lp-Space (M,k)) st f in x holds
( x = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) )
let k be positive Real; ::_thesis: for x being Point of (Lp-Space (M,k)) st f in x holds
( x = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) )
let x be Point of (Lp-Space (M,k)); ::_thesis: ( f in x implies ( x = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) ) )
assume A1: f in x ; ::_thesis: ( x = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) )
 x in the carrier of (Pre-Lp-Space (M,k)) ;
then  x in CosetSet (M,k) by Def11;
then consider g being PartFunc of X,REAL such that
A2: ( x = a.e-eq-class_Lp (g,M,k) & g in Lp_Functions (M,k) ) ;
 g in x by A2, Th38;
then  ( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) by A1, Th50;
hence  ( x = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) ) by Th53, A1, A2, Th42; ::_thesis: verum
end;

theorem Th56: :: LPSPACE2:56
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds X --> 0 in L1_Functions M
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S holds X --> 0 in L1_Functions M
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S holds X --> 0 in L1_Functions M
let M be sigma_Measure of S; ::_thesis: X --> 0 in L1_Functions M
reconsider ND = {} as Element of S by MEASURE1:34;
A1: M . ND = 0 by VALUED_0:def_19;
 X --> 0 is Function of X,REAL by FUNCOP_1:46;
then A2: dom (X --> 0) = ND ` by FUNCT_2:def_1;
 for x being Element of X st x in dom (X --> 0) holds
(X --> 0) . x = 0 by FUNCOP_1:7;
then  X --> 0 is_integrable_on M by A2, Th15;
hence  X --> 0 in L1_Functions M by A1, A2; ::_thesis: verum
end;

theorem Th57: :: LPSPACE2:57
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & Integral (M,((abs f) to_power k)) = 0 holds
f a.e.= X --> 0,M
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & Integral (M,((abs f) to_power k)) = 0 holds
f a.e.= X --> 0,M
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & Integral (M,((abs f) to_power k)) = 0 holds
f a.e.= X --> 0,M
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) & Integral (M,((abs f) to_power k)) = 0 holds
f a.e.= X --> 0,M
let f be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) & Integral (M,((abs f) to_power k)) = 0 holds
f a.e.= X --> 0,M
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) & Integral (M,((abs f) to_power k)) = 0 implies f a.e.= X --> 0,M )
assume that
A1: f in Lp_Functions (M,k) and
A2: Integral (M,((abs f) to_power k)) = 0 ; ::_thesis: f a.e.= X --> 0,M
 ex h being PartFunc of X,REAL st
( f = h & ex ND being Element of S st
( M . (ND `) = 0 & dom h = ND & h is_measurable_on ND & (abs h) to_power k is_integrable_on M ) ) by A1;
then consider NDf being Element of S such that
A3: ( M . (NDf `) = 0 & dom f = NDf & f is_measurable_on NDf & (abs f) to_power k is_integrable_on M ) ;
reconsider t = (abs f) to_power k as PartFunc of X,REAL ;
reconsider ND = NDf ` as Element of S by MEASURE1:34;
A4: dom t = dom (abs f) by MESFUN6C:def_4;
then A5: dom t = NDf by A3, VALUED_1:def_11;
 dom t = ND ` by A4, A3, VALUED_1:def_11;
then A6: t in L1_Functions M by A3;
 abs t = t by Th14;
then  t a.e.= X --> 0,M by A2, A6, LPSPACE1:53;
then consider ND1 being Element of S such that
A7: ( M . ND1 = 0 & ((abs f) to_power k) | (ND1 `) = (X --> 0) | (ND1 `) ) by LPSPACE1:def_10;
set ND2 = ND \/ ND1;
 ( ND is measure_zero of M & ND1 is measure_zero of M ) by A3, A7, MEASURE1:def_7;
then  ND \/ ND1 is measure_zero of M by MEASURE1:37;
then A8: M . (ND \/ ND1) = 0 by MEASURE1:def_7;
A9: ( (ND \/ ND1) ` c= ND ` & (ND \/ ND1) ` c= ND1 ` ) by XBOOLE_1:7, XBOOLE_1:34;
 dom (X --> 0) = X by FUNCOP_1:13;
then A10: dom ((X --> 0) | ((ND \/ ND1) `)) = (ND \/ ND1) ` by RELAT_1:62;
A11: dom (f | ((ND \/ ND1) `)) = (ND \/ ND1) ` by A3, A9, RELAT_1:62;
 for x being set st x in dom (f | ((ND \/ ND1) `)) holds
(f | ((ND \/ ND1) `)) . x = ((X --> 0) | ((ND \/ ND1) `)) . x
proof
let x be set ; ::_thesis: ( x in dom (f | ((ND \/ ND1) `)) implies (f | ((ND \/ ND1) `)) . x = ((X --> 0) | ((ND \/ ND1) `)) . x )
assume A12: x in dom (f | ((ND \/ ND1) `)) ; ::_thesis: (f | ((ND \/ ND1) `)) . x = ((X --> 0) | ((ND \/ ND1) `)) . x
A13: now__::_thesis:_not_f_._x_<>_0
assume  f . x <> 0 ; ::_thesis: contradiction
then  abs (f . x) > 0 by COMPLEX1:47;
then  (abs (f . x)) to_power k <> 0 by POWER:34;
then  ((abs f) . x) to_power k <> 0 by VALUED_1:18;
then A14: ((abs f) to_power k) . x <> 0 by A5, A9, A12, A11, MESFUN6C:def_4;
 ((X --> 0) | (ND1 `)) . x = (X --> 0) . x by A9, A12, A11, FUNCT_1:49;
then  ((X --> 0) | (ND1 `)) . x = 0 by A12, FUNCOP_1:7;
hence  contradiction by A14, A7, A9, A12, A11, FUNCT_1:49; ::_thesis: verum
end;
 ((X --> 0) | ((ND \/ ND1) `)) . x = (X --> 0) . x by A11, A12, FUNCT_1:49;
then  ((X --> 0) | ((ND \/ ND1) `)) . x = 0 by A12, FUNCOP_1:7;
hence  (f | ((ND \/ ND1) `)) . x = ((X --> 0) | ((ND \/ ND1) `)) . x by A11, A12, A13, FUNCT_1:49; ::_thesis: verum
end;
then  f | ((ND \/ ND1) `) = (X --> 0) | ((ND \/ ND1) `) by A10, A11, FUNCT_1:def_11;
hence  f a.e.= X --> 0,M by A8, LPSPACE1:def_10; ::_thesis: verum
end;

theorem Th58: :: LPSPACE2:58
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds Integral (M,((abs (X --> 0)) to_power k)) = 0
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds Integral (M,((abs (X --> 0)) to_power k)) = 0
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real holds Integral (M,((abs (X --> 0)) to_power k)) = 0
let M be sigma_Measure of S; ::_thesis: for k being positive Real holds Integral (M,((abs (X --> 0)) to_power k)) = 0
let k be positive Real; ::_thesis: Integral (M,((abs (X --> 0)) to_power k)) = 0
A1: for x being set st x in dom (X --> 0) holds
0 <= (X --> 0) . x ;
then  Integral (M,((abs (X --> 0)) to_power k)) = Integral (M,((X --> 0) to_power k)) by Th14, MESFUNC6:52
.= Integral (M,(X --> 0)) by Th12
.= Integral (M,(abs (X --> 0))) by A1, Th14, MESFUNC6:52 ;
hence  Integral (M,((abs (X --> 0)) to_power k)) = 0 by LPSPACE1:54; ::_thesis: verum
end;

theorem Th59: :: LPSPACE2:59
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
( f (#) g in L1_Functions M & f (#) g is_integrable_on M )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
( f (#) g in L1_Functions M & f (#) g is_integrable_on M )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
( f (#) g in L1_Functions M & f (#) g is_integrable_on M )
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
( f (#) g in L1_Functions M & f (#) g is_integrable_on M )
let f, g be PartFunc of X,REAL; ::_thesis: for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
( f (#) g in L1_Functions M & f (#) g is_integrable_on M )
let m, n be positive Real; ::_thesis: ( (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) implies ( f (#) g in L1_Functions M & f (#) g is_integrable_on M ) )
assume that
A1: (1 / m) + (1 / n) = 1 and
A2: ( f in Lp_Functions (M,m) & g in Lp_Functions (M,n) ) ; ::_thesis: ( f (#) g in L1_Functions M & f (#) g is_integrable_on M )
A3: ( m > 1 & n > 1 ) by A1, Th1;
consider f1 being PartFunc of X,REAL such that
A4: ( f = f1 & ex NDf being Element of S st
( M . (NDf `) = 0 & dom f1 = NDf & f1 is_measurable_on NDf & (abs f1) to_power m is_integrable_on M ) ) by A2;
consider EDf being Element of S such that
A5: ( M . (EDf `) = 0 & dom f1 = EDf & f1 is_measurable_on EDf ) by A4;
consider g1 being PartFunc of X,REAL such that
A6: ( g = g1 & ex NDg being Element of S st
( M . (NDg `) = 0 & dom g1 = NDg & g1 is_measurable_on NDg & (abs g1) to_power n is_integrable_on M ) ) by A2;
consider EDg being Element of S such that
A7: ( M . (EDg `) = 0 & dom g1 = EDg & g1 is_measurable_on EDg ) by A6;
set u = (abs f1) to_power m;
set v = (abs g1) to_power n;
set w = f1 (#) g1;
set z = ((1 / m) (#) ((abs f1) to_power m)) + ((1 / n) (#) ((abs g1) to_power n));
A8: ( dom f1 = dom (abs f1) & dom g1 = dom (abs g1) ) by VALUED_1:def_11;
then A9: ( dom ((abs f1) to_power m) = dom f1 & dom ((abs g1) to_power n) = dom g1 ) by MESFUN6C:def_4;
then A10: dom (f1 (#) g1) = (dom ((abs f1) to_power m)) /\ (dom ((abs g1) to_power n)) by VALUED_1:def_4;
set Nf = EDf ` ;
set Ng = EDg ` ;
set E = EDf /\ EDg;
reconsider Nf = EDf ` , Ng = EDg ` as Element of S by MEASURE1:34;
 ( dom ((abs f1) to_power m) = Nf ` & dom ((abs g1) to_power n) = Ng ` ) by A5, A7, A8, MESFUN6C:def_4;
then  ( (abs f1) to_power m in L1_Functions M & (abs g1) to_power n in L1_Functions M ) by A4, A5, A6, A7;
then  ( (1 / m) (#) ((abs f1) to_power m) in L1_Functions M & (1 / n) (#) ((abs g1) to_power n) in L1_Functions M ) by LPSPACE1:24;
then  ((1 / m) (#) ((abs f1) to_power m)) + ((1 / n) (#) ((abs g1) to_power n)) in L1_Functions M by LPSPACE1:23;
then A11: ex h being PartFunc of X,REAL st
( ((1 / m) (#) ((abs f1) to_power m)) + ((1 / n) (#) ((abs g1) to_power n)) = h & ex ND being Element of S st
( M . ND = 0 & dom h = ND ` & h is_integrable_on M ) ) ;
 ( dom ((1 / m) (#) ((abs f1) to_power m)) = dom ((abs f1) to_power m) & dom ((1 / n) (#) ((abs g1) to_power n)) = dom ((abs g1) to_power n) ) by VALUED_1:def_5;
then A12: dom (((1 / m) (#) ((abs f1) to_power m)) + ((1 / n) (#) ((abs g1) to_power n))) = (dom ((abs f1) to_power m)) /\ (dom ((abs g1) to_power n)) by VALUED_1:def_1;
A13: (EDf /\ EDg) ` = (EDf `) \/ (EDg `) by XBOOLE_1:54;
 ( Nf is measure_zero of M & Ng is measure_zero of M ) by A5, A7, MEASURE1:def_7;
then  Nf \/ Ng is measure_zero of M by MEASURE1:37;
then A14: M . ((EDf /\ EDg) `) = 0 by A13, MEASURE1:def_7;
 ( f1 is_measurable_on EDf /\ EDg & g1 is_measurable_on EDf /\ EDg ) by A5, A7, MESFUNC6:16, XBOOLE_1:17;
then A15: f1 (#) g1 is_measurable_on EDf /\ EDg by A5, A7, MESFUN7C:31;
 for x being Element of X st x in dom (f1 (#) g1) holds
abs ((f1 (#) g1) . x) <= (((1 / m) (#) ((abs f1) to_power m)) + ((1 / n) (#) ((abs g1) to_power n))) . x
proof
let x be Element of X; ::_thesis: ( x in dom (f1 (#) g1) implies abs ((f1 (#) g1) . x) <= (((1 / m) (#) ((abs f1) to_power m)) + ((1 / n) (#) ((abs g1) to_power n))) . x )
assume A16: x in dom (f1 (#) g1) ; ::_thesis: abs ((f1 (#) g1) . x) <= (((1 / m) (#) ((abs f1) to_power m)) + ((1 / n) (#) ((abs g1) to_power n))) . x
 abs (f1 (#) g1) = (abs f1) (#) (abs g1) by RFUNCT_1:24;
then  (abs (f1 (#) g1)) . x = ((abs f1) . x) * ((abs g1) . x) by VALUED_1:5;
then A17: abs ((f1 (#) g1) . x) = ((abs f1) . x) * ((abs g1) . x) by VALUED_1:18;
A18: ( (abs f1) . x >= 0 & (abs g1) . x >= 0 ) by MESFUNC6:51;
 ( x in dom ((abs f1) to_power m) & x in dom ((abs g1) to_power n) ) by A16, A10, XBOOLE_0:def_4;
then  ( (((abs f1) . x) to_power m) / m = (1 / m) * (((abs f1) to_power m) . x) & (((abs g1) . x) to_power n) / n = (1 / n) * (((abs g1) to_power n) . x) ) by MESFUN6C:def_4;
then  ( (((abs f1) . x) to_power m) / m = ((1 / m) (#) ((abs f1) to_power m)) . x & (((abs g1) . x) to_power n) / n = ((1 / n) (#) ((abs g1) to_power n)) . x ) by VALUED_1:6;
then  abs ((f1 (#) g1) . x) <= (((1 / m) (#) ((abs f1) to_power m)) . x) + (((1 / n) (#) ((abs g1) to_power n)) . x) by A1, A3, A17, A18, HOLDER_1:5;
hence  abs ((f1 (#) g1) . x) <= (((1 / m) (#) ((abs f1) to_power m)) + ((1 / n) (#) ((abs g1) to_power n))) . x by A16, A10, A12, VALUED_1:def_1; ::_thesis: verum
end;
then A19: f1 (#) g1 is_integrable_on M by A5, A7, A9, A10, A11, A15, A12, MESFUNC6:96;
set ND = (EDf /\ EDg) ` ;
reconsider ND = (EDf /\ EDg) ` as Element of S by MEASURE1:34;
 dom (f1 (#) g1) = ND ` by A5, A7, VALUED_1:def_4;
hence  ( f (#) g in L1_Functions M & f (#) g is_integrable_on M ) by A4, A6, A14, A19; ::_thesis: verum
end;

theorem Th60: :: LPSPACE2:60
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) )
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) )
let f, g be PartFunc of X,REAL; ::_thesis: for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) holds
ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) )
let m, n be positive Real; ::_thesis: ( (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) implies ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) ) )
assume A1: ( (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,n) ) ; ::_thesis: ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) )
then A2: ( m > 1 & n > 1 ) by Th1;
consider f1 being PartFunc of X,REAL such that
A3: ( f = f1 & ex NDf being Element of S st
( M . (NDf `) = 0 & dom f1 = NDf & f1 is_measurable_on NDf & (abs f1) to_power m is_integrable_on M ) ) by A1;
consider EDf being Element of S such that
A4: ( M . (EDf `) = 0 & dom f1 = EDf & f1 is_measurable_on EDf ) by A3;
consider g1 being PartFunc of X,REAL such that
A5: ( g = g1 & ex NDg being Element of S st
( M . (NDg `) = 0 & dom g1 = NDg & g1 is_measurable_on NDg & (abs g1) to_power n is_integrable_on M ) ) by A1;
consider EDg being Element of S such that
A6: ( M . (EDg `) = 0 & dom g1 = EDg & g1 is_measurable_on EDg ) by A5;
set u = (abs f1) to_power m;
set v = (abs g1) to_power n;
A7: ( 0 <= Integral (M,((abs f1) to_power m)) & 0 <= Integral (M,((abs g1) to_power n)) ) by A3, A5, A1, Th49;
reconsider s1 = Integral (M,((abs f1) to_power m)), s2 = Integral (M,((abs g1) to_power n)) as Real by A3, A5, A1, Th49;
A8: ( dom f1 = dom (abs f1) & dom g1 = dom (abs g1) ) by VALUED_1:def_11;
reconsider Nf = EDf ` , Ng = EDg ` as Element of S by MEASURE1:34;
set t1 = s1 to_power (1 / m);
set t2 = s2 to_power (1 / n);
set E = EDf /\ EDg;
A9: (EDf /\ EDg) ` = (EDf `) \/ (EDg `) by XBOOLE_1:54;
 ( Nf is measure_zero of M & Ng is measure_zero of M ) by A4, A6, MEASURE1:def_7;
then A10: (EDf /\ EDg) ` is measure_zero of M by A9, MEASURE1:37;
A11: dom (f1 (#) g1) = EDf /\ EDg by A4, A6, VALUED_1:def_4;
 ( f1 is_measurable_on EDf /\ EDg & g1 is_measurable_on EDf /\ EDg ) by A4, A6, MESFUNC6:16, XBOOLE_1:17;
then A12: f1 (#) g1 is_measurable_on EDf /\ EDg by A4, A6, MESFUN7C:31;
A13: f1 (#) g1 in L1_Functions M by A1, A3, A5, Th59;
then A14: ex fg1 being PartFunc of X,REAL st
( fg1 = f1 (#) g1 & ex ND being Element of S st
( M . ND = 0 & dom fg1 = ND ` & fg1 is_integrable_on M ) ) ;
then A15: ( Integral (M,(abs (f1 (#) g1))) in REAL & abs (f1 (#) g1) is_integrable_on M ) by LPSPACE1:44;
percases ( ( s1 = 0 & s2 >= 0 ) or ( s1 > 0 & s2 = 0 ) or ( s1 <> 0 & s2 <> 0 ) ) by A3, A5, A1, Th49;
supposeA16: ( s1 = 0 & s2 >= 0 ) ; ::_thesis: ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) )
 f1 in Lp_Functions (M,m) by A3;
then  f1 a.e.= X --> 0,M by A16, Th57;
then consider Nf1 being Element of S such that
A17: ( M . Nf1 = 0 & f1 | (Nf1 `) = (X --> 0) | (Nf1 `) ) by LPSPACE1:def_10;
reconsider Z = ((EDf /\ EDg) \ Nf1) ` as Element of S by MEASURE1:34;
A18: ((EDf /\ EDg) \ Nf1) ` = ((EDf /\ EDg) `) \/ Nf1 by SUBSET_1:14;
 Nf1 is measure_zero of M by A17, MEASURE1:def_7;
then  Z is measure_zero of M by A10, A18, MEASURE1:37;
then A19: M . Z = 0 by MEASURE1:def_7;
 dom (X --> 0) = X by FUNCOP_1:13;
then A20: dom ((X --> 0) | (Z `)) = Z ` by RELAT_1:62;
A21: dom ((f1 (#) g1) | (Z `)) = Z ` by A11, RELAT_1:62, XBOOLE_1:36;
 for x being set st x in dom ((f1 (#) g1) | (Z `)) holds
((f1 (#) g1) | (Z `)) . x = ((X --> 0) | (Z `)) . x
proof
let x be set ; ::_thesis: ( x in dom ((f1 (#) g1) | (Z `)) implies ((f1 (#) g1) | (Z `)) . x = ((X --> 0) | (Z `)) . x )
assume A22: x in dom ((f1 (#) g1) | (Z `)) ; ::_thesis: ((f1 (#) g1) | (Z `)) . x = ((X --> 0) | (Z `)) . x
then  ( x in X & not x in Nf1 ) by A21, XBOOLE_0:def_5;
then  x in Nf1 ` by XBOOLE_0:def_5;
then  ( f1 . x = (f1 | (Nf1 `)) . x & (X --> 0) . x = ((X --> 0) | (Nf1 `)) . x ) by FUNCT_1:49;
then A23: f1 . x = 0 by A17, A22, FUNCOP_1:7;
A24: dom ((f1 (#) g1) | (Z `)) c= dom (f1 (#) g1) by RELAT_1:60;
((f1 (#) g1) | (Z `)) . x = (f1 (#) g1) . x by A22, FUNCT_1:47
.= (f1 . x) * (g1 . x) by A22, A24, VALUED_1:def_4
.= ((Z `) --> 0) . x by A22, A21, A23, FUNCOP_1:7
.= ((X /\ (Z `)) --> 0) . x by XBOOLE_1:28 ;
hence  ((f1 (#) g1) | (Z `)) . x = ((X --> 0) | (Z `)) . x by FUNCOP_1:12; ::_thesis: verum
end;
then  (f1 (#) g1) | (Z `) = (X --> 0) | (Z `) by A20, A21, FUNCT_1:def_11;
then A25: f1 (#) g1 a.e.= X --> 0,M by A19, LPSPACE1:def_10;
 X --> 0 in L1_Functions M by Th56;
then  Integral (M,(abs (f1 (#) g1))) = Integral (M,(abs (X --> 0))) by A13, A25, LPSPACE1:45;
then A26: Integral (M,(abs (f1 (#) g1))) = 0 by LPSPACE1:54;
 (s1 to_power (1 / m)) * (s2 to_power (1 / n)) = 0 * (s2 to_power (1 / n)) by A16, POWER:def_2;
hence  ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) ) by A3, A5, A26; ::_thesis: verum
end;
supposeA27: ( s1 > 0 & s2 = 0 ) ; ::_thesis: ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) )
 g1 in Lp_Functions (M,n) by A5;
then  g1 a.e.= X --> 0,M by A27, Th57;
then consider Ng1 being Element of S such that
A28: ( M . Ng1 = 0 & g1 | (Ng1 `) = (X --> 0) | (Ng1 `) ) by LPSPACE1:def_10;
reconsider Z = ((EDf /\ EDg) \ Ng1) ` as Element of S by MEASURE1:34;
A29: ((EDf /\ EDg) \ Ng1) ` = ((EDf /\ EDg) `) \/ Ng1 by SUBSET_1:14;
 Ng1 is measure_zero of M by A28, MEASURE1:def_7;
then  Z is measure_zero of M by A10, A29, MEASURE1:37;
then A30: M . Z = 0 by MEASURE1:def_7;
 dom (X --> 0) = X by FUNCOP_1:13;
then A31: dom ((X --> 0) | (Z `)) = Z ` by RELAT_1:62;
A32: dom ((f1 (#) g1) | (Z `)) = Z ` by A11, RELAT_1:62, XBOOLE_1:36;
 for x being set st x in dom ((f1 (#) g1) | (Z `)) holds
((f1 (#) g1) | (Z `)) . x = ((X --> 0) | (Z `)) . x
proof
let x be set ; ::_thesis: ( x in dom ((f1 (#) g1) | (Z `)) implies ((f1 (#) g1) | (Z `)) . x = ((X --> 0) | (Z `)) . x )
assume A33: x in dom ((f1 (#) g1) | (Z `)) ; ::_thesis: ((f1 (#) g1) | (Z `)) . x = ((X --> 0) | (Z `)) . x
then  ( x in X & not x in Ng1 ) by A32, XBOOLE_0:def_5;
then  x in Ng1 ` by XBOOLE_0:def_5;
then  ( g1 . x = (g1 | (Ng1 `)) . x & (X --> 0) . x = ((X --> 0) | (Ng1 `)) . x ) by FUNCT_1:49;
then A34: g1 . x = 0 by A28, A33, FUNCOP_1:7;
A35: dom ((f1 (#) g1) | (Z `)) c= dom (f1 (#) g1) by RELAT_1:60;
((f1 (#) g1) | (Z `)) . x = (f1 (#) g1) . x by A33, FUNCT_1:47
.= (f1 . x) * (g1 . x) by A33, A35, VALUED_1:def_4
.= ((Z `) --> 0) . x by A33, A32, A34, FUNCOP_1:7
.= ((X /\ (Z `)) --> 0) . x by XBOOLE_1:28 ;
hence  ((f1 (#) g1) | (Z `)) . x = ((X --> 0) | (Z `)) . x by FUNCOP_1:12; ::_thesis: verum
end;
then  (f1 (#) g1) | (Z `) = (X --> 0) | (Z `) by A31, A32, FUNCT_1:def_11;
then A36: f1 (#) g1 a.e.= X --> 0,M by A30, LPSPACE1:def_10;
 X --> 0 in L1_Functions M by Th56;
then  Integral (M,(abs (f1 (#) g1))) = Integral (M,(abs (X --> 0))) by A13, A36, LPSPACE1:45;
then A37: Integral (M,(abs (f1 (#) g1))) = 0 by LPSPACE1:54;
 (s1 to_power (1 / m)) * (s2 to_power (1 / n)) = (s1 to_power (1 / m)) * 0 by A27, POWER:def_2;
hence  ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) ) by A3, A5, A37; ::_thesis: verum
end;
supposeA38: ( s1 <> 0 & s2 <> 0 ) ; ::_thesis: ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) )
then A39: ( s1 to_power (1 / m) > 0 & s2 to_power (1 / n) > 0 ) by A7, POWER:34;
then A40: abs (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) = 1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n))) by ABSVALUE:def_1;
set w = (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1);
set F = (1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m);
set G = (1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n);
set z = ((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n));
A41: ( dom ((1 / (s1 to_power (1 / m))) (#) (abs f1)) = dom (abs f1) & dom ((1 / (s2 to_power (1 / n))) (#) (abs g1)) = dom (abs g1) ) by VALUED_1:def_5;
 ( dom ((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) = dom (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m) & dom ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)) = dom (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n) ) by VALUED_1:def_5;
then A42: ( dom ((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) = dom (abs f1) & dom ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)) = dom (abs g1) ) by A41, MESFUN6C:def_4;
then A43: dom (((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) = (dom (abs f1)) /\ (dom (abs g1)) by VALUED_1:def_1;
 ( ((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m = ((1 / (s1 to_power (1 / m))) to_power m) (#) ((abs f1) to_power m) & ((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n = ((1 / (s2 to_power (1 / n))) to_power n) (#) ((abs g1) to_power n) ) by A39, Th19;
then A44: ( ((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m is_integrable_on M & ((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n is_integrable_on M ) by A3, A5, MESFUNC6:102;
then A45: ( (1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m) is_integrable_on M & (1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n) is_integrable_on M ) by MESFUNC6:102;
then A46: ((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)) is_integrable_on M by MESFUNC6:100;
A47: dom ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) = dom (f1 (#) g1) by VALUED_1:def_5;
then A48: dom ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) = (dom f1) /\ (dom g1) by VALUED_1:def_4;
 dom ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (abs (f1 (#) g1))) = dom (abs (f1 (#) g1)) by VALUED_1:def_5;
then A49: dom ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (abs (f1 (#) g1))) = dom (f1 (#) g1) by VALUED_1:def_11;
A50: (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1) is_measurable_on EDf /\ EDg by A11, A12, MESFUNC6:21;
 for x being Element of X st x in dom ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) holds
abs (((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) . x) <= (((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) . x
proof
let x be Element of X; ::_thesis: ( x in dom ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) implies abs (((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) . x) <= (((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) . x )
assume A51: x in dom ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) ; ::_thesis: abs (((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) . x) <= (((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) . x
 ( (abs f1) . x >= 0 & (abs g1) . x >= 0 ) by MESFUNC6:51;
then A52: ((1 / (s1 to_power (1 / m))) * ((abs f1) . x)) * ((1 / (s2 to_power (1 / n))) * ((abs g1) . x)) <= ((((1 / (s1 to_power (1 / m))) * ((abs f1) . x)) to_power m) / m) + ((((1 / (s2 to_power (1 / n))) * ((abs g1) . x)) to_power n) / n) by A1, A2, A39, HOLDER_1:5;
 dom ((abs f1) (#) (abs g1)) = (dom (abs f1)) /\ (dom (abs g1)) by VALUED_1:def_4;
then A53: ((abs f1) (#) (abs g1)) . x = ((abs f1) . x) * ((abs g1) . x) by A8, A48, A51, VALUED_1:def_4;
A54: ((1 / (s1 to_power (1 / m))) * ((abs f1) . x)) * ((1 / (s2 to_power (1 / n))) * ((abs g1) . x)) = (((1 / (s1 to_power (1 / m))) * (1 / (s2 to_power (1 / n)))) * ((abs f1) . x)) * ((abs g1) . x)
.= ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) * ((abs f1) . x)) * ((abs g1) . x) by XCMPLX_1:102
.= (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) * (((abs f1) (#) (abs g1)) . x) by A53
.= (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) * ((abs (f1 (#) g1)) . x) by RFUNCT_1:24
.= ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (abs (f1 (#) g1))) . x by A47, A51, A49, VALUED_1:def_5
.= (abs ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1))) . x by A40, RFUNCT_1:25 ;
A55: ( (1 / (s1 to_power (1 / m))) * ((abs f1) . x) = ((1 / (s1 to_power (1 / m))) (#) (abs f1)) . x & (1 / (s2 to_power (1 / n))) * ((abs g1) . x) = ((1 / (s2 to_power (1 / n))) (#) (abs g1)) . x ) by VALUED_1:6;
 ( dom (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m) = dom f1 & dom (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n) = dom g1 ) by A8, A41, MESFUN6C:def_4;
then  ( x in dom (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m) & x in dom (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n) ) by A48, A51, XBOOLE_0:def_4;
then  ( (((1 / (s1 to_power (1 / m))) (#) (abs f1)) . x) to_power m = (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m) . x & (((1 / (s2 to_power (1 / n))) (#) (abs g1)) . x) to_power n = (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n) . x ) by MESFUN6C:def_4;
then  ( ((((1 / (s1 to_power (1 / m))) (#) (abs f1)) . x) to_power m) / m = ((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) . x & ((((1 / (s2 to_power (1 / n))) (#) (abs g1)) . x) to_power n) / n = ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)) . x ) by VALUED_1:6;
then  ((((1 / (s1 to_power (1 / m))) * ((abs f1) . x)) to_power m) / m) + ((((1 / (s2 to_power (1 / n))) * ((abs g1) . x)) to_power n) / n) = (((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) . x by A8, A48, A51, A43, A55, VALUED_1:def_1;
hence  abs (((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) . x) <= (((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) . x by A52, A54, VALUED_1:18; ::_thesis: verum
end;
then A56: Integral (M,(abs ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)))) <= Integral (M,(((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)))) by A4, A6, A46, A8, A48, A43, A50, MESFUNC6:96;
A57: ex E1 being Element of S st
( E1 = (dom ((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m))) /\ (dom ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) & Integral (M,(((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)))) = (Integral (M,(((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) | E1))) + (Integral (M,(((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)) | E1))) ) by A45, MESFUNC6:101;
 ( EDf = X /\ EDf & EDg = X /\ EDg ) by XBOOLE_1:28;
then A58: ( EDf = X \ Nf & EDg = X \ Ng ) by XBOOLE_1:48;
A59: EDf \ (EDf /\ EDg) = EDf \ EDg by XBOOLE_1:47
.= ((X \ Nf) \ X) \/ ((X \ Nf) /\ Ng) by A58, XBOOLE_1:52
.= (X \ (Nf \/ X)) \/ ((X \ Nf) /\ Ng) by XBOOLE_1:41
.= (X \ X) \/ ((X \ Nf) /\ Ng) by XBOOLE_1:12
.= {} \/ ((X \ Nf) /\ Ng) by XBOOLE_1:37 ;
A60: EDg \ (EDf /\ EDg) = EDg \ EDf by XBOOLE_1:47
.= ((X \ Ng) \ X) \/ ((X \ Ng) /\ Nf) by A58, XBOOLE_1:52
.= (X \ (Ng \/ X)) \/ ((X \ Ng) /\ Nf) by XBOOLE_1:41
.= (X \ X) \/ ((X \ Ng) /\ Nf) by XBOOLE_1:12
.= {} \/ ((X \ Ng) /\ Nf) by XBOOLE_1:37 ;
set NF = EDf /\ Ng;
set NG = EDg /\ Nf;
 ( Nf is measure_zero of M & Ng is measure_zero of M ) by A4, A6, MEASURE1:def_7;
then  ( EDf /\ Ng is measure_zero of M & EDg /\ Nf is measure_zero of M ) by MEASURE1:36, XBOOLE_1:17;
then A61: ( M . (EDf /\ Ng) = 0 & M . (EDg /\ Nf) = 0 ) by MEASURE1:def_7;
 ( EDf /\ EDg = EDf /\ (EDf /\ EDg) & EDf /\ EDg = EDg /\ (EDf /\ EDg) ) by XBOOLE_1:17, XBOOLE_1:28;
then A62: ( EDf /\ EDg = EDf \ (EDf /\ Ng) & EDf /\ EDg = EDg \ (EDg /\ Nf) ) by A58, A59, A60, XBOOLE_1:48;
 R_EAL ((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) is_integrable_on M by A45, MESFUNC6:def_4;
then  ex E being Element of S st
( E = dom (R_EAL ((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m))) & R_EAL ((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) is_measurable_on E ) by MESFUNC5:def_17;
then A63: (1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m) is_measurable_on EDf by A42, A8, A4, MESFUNC6:def_1;
 R_EAL ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)) is_integrable_on M by A45, MESFUNC6:def_4;
then  ex E being Element of S st
( E = dom (R_EAL ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) & R_EAL ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)) is_measurable_on E ) by MESFUNC5:def_17;
then A64: (1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n) is_measurable_on EDg by A42, A8, A6, MESFUNC6:def_1;
 (1 / (s1 to_power (1 / m))) to_power m = (s1 to_power (1 / m)) to_power (- m) by A38, A7, POWER:32, POWER:34;
then  (1 / (s1 to_power (1 / m))) to_power m = s1 to_power ((1 / m) * (- m)) by A7, A38, POWER:33;
then  (1 / (s1 to_power (1 / m))) to_power m = s1 to_power (- ((1 * (1 / m)) * m)) ;
then  (1 / (s1 to_power (1 / m))) to_power m = s1 to_power (- 1) by XCMPLX_1:106;
then  (1 / (s1 to_power (1 / m))) to_power m = (1 / s1) to_power 1 by A7, A38, POWER:32;
then A65: (1 / (s1 to_power (1 / m))) to_power m = 1 / s1 by POWER:25;
 ( (R_EAL (1 / s1)) * (R_EAL s1) = (1 / s1) * s1 & (R_EAL (1 / s2)) * (R_EAL s2) = (1 / s2) * s2 ) by EXTREAL1:5;
then A66: ( (R_EAL (1 / s1)) * (R_EAL s1) = 1 & (R_EAL (1 / s2)) * (R_EAL s2) = 1 ) by A38, XCMPLX_1:106;
A67: (1 / (s2 to_power (1 / n))) to_power n = (s2 to_power (1 / n)) to_power (- n) by A38, A7, POWER:32, POWER:34
.= s2 to_power ((1 / n) * (- n)) by A7, A38, POWER:33
.= s2 to_power (- ((1 * (1 / n)) * n))
.= s2 to_power (- 1) by XCMPLX_1:106
.= (1 / s2) to_power 1 by A7, A38, POWER:32
.= 1 / s2 by POWER:25 ;
A68: Integral (M,(((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) | (EDf /\ EDg))) = Integral (M,((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m))) by A4, A8, A42, A62, A61, A63, MESFUNC6:89
.= (R_EAL (1 / m)) * (Integral (M,(((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m))) by A44, MESFUNC6:102
.= (R_EAL (1 / m)) * (Integral (M,(((1 / (s1 to_power (1 / m))) to_power m) (#) ((abs f1) to_power m)))) by A39, Th19
.= (R_EAL (1 / m)) * ((R_EAL ((1 / (s1 to_power (1 / m))) to_power m)) * (Integral (M,((abs f1) to_power m)))) by A3, MESFUNC6:102
.= 1 / m by A65, A66, XXREAL_3:81 ;
Integral (M,(((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)) | (EDf /\ EDg))) = Integral (M,((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) by A6, A8, A42, A62, A61, A64, MESFUNC6:89
.= (R_EAL (1 / n)) * (Integral (M,(((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n))) by A44, MESFUNC6:102
.= (R_EAL (1 / n)) * (Integral (M,(((1 / (s2 to_power (1 / n))) to_power n) (#) ((abs g1) to_power n)))) by A39, Th19
.= (R_EAL (1 / n)) * ((R_EAL ((1 / (s2 to_power (1 / n))) to_power n)) * (Integral (M,((abs g1) to_power n)))) by A5, MESFUNC6:102
.= 1 / n by A66, A67, XXREAL_3:81 ;
then A69: Integral (M,(((1 / m) (#) (((1 / (s1 to_power (1 / m))) (#) (abs f1)) to_power m)) + ((1 / n) (#) (((1 / (s2 to_power (1 / n))) (#) (abs g1)) to_power n)))) = 1 by A1, A42, A4, A6, A8, A57, A68, SUPINF_2:1;
 abs ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) = (abs (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n))))) (#) (abs (f1 (#) g1)) by RFUNCT_1:25;
then  abs ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)) = (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (abs (f1 (#) g1)) by A39, ABSVALUE:def_1;
then A70: Integral (M,(abs ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) (#) (f1 (#) g1)))) = (R_EAL (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n))))) * (Integral (M,(abs (f1 (#) g1)))) by A15, MESFUNC6:102;
reconsider c1 = Integral (M,(abs (f1 (#) g1))) as Real by A14, LPSPACE1:44;
 (R_EAL (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n))))) * (Integral (M,(abs (f1 (#) g1)))) = (R_EAL (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n))))) * (R_EAL c1) ;
then  (R_EAL (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n))))) * (Integral (M,(abs (f1 (#) g1)))) = (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) * c1 by EXTREAL1:5;
then  ((s1 to_power (1 / m)) * (s2 to_power (1 / n))) * ((1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) * c1) <= ((s1 to_power (1 / m)) * (s2 to_power (1 / n))) * 1 by A39, A56, A70, A69, XREAL_1:64;
then A71: (((s1 to_power (1 / m)) * (s2 to_power (1 / n))) * (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n))))) * c1 <= (s1 to_power (1 / m)) * (s2 to_power (1 / n)) ;
 ((s1 to_power (1 / m)) * (s2 to_power (1 / n))) * (1 / ((s1 to_power (1 / m)) * (s2 to_power (1 / n)))) = 1 by A39, XCMPLX_1:106;
hence  ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power n)) & Integral (M,(abs (f (#) g))) <= (r1 to_power (1 / m)) * (r2 to_power (1 / n)) ) ) by A3, A5, A71; ::_thesis: verum
end;
end;
end;

Lm5:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) holds
ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) holds
ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) holds
ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) )
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) holds
ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) )
let f, g be PartFunc of X,REAL; ::_thesis: for m, n being positive Real st (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) holds
ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) )
let m, n be positive Real; ::_thesis: ( (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) implies ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) ) )
assume A1: ( (1 / m) + (1 / n) = 1 & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) ) ; ::_thesis: ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) )
then  ((m + n) * ((m * n) ")) * (m * n) = 1 * (m * n) by XCMPLX_1:211;
then  (m + n) * (((m * n) ") * (m * n)) = m * n ;
then  (m + n) * 1 = m * n by XCMPLX_0:def_7;
then A2: m = n * (m - 1) ;
A3: 1 - 1 < m - 1 by A1, Th1, XREAL_1:14;
then A4: m - 1 > 0 ;
 ex f1 being PartFunc of X,REAL st
( f = f1 & ex NDf being Element of S st
( M . (NDf `) = 0 & dom f1 = NDf & f1 is_measurable_on NDf & (abs f1) to_power m is_integrable_on M ) ) by A1;
then consider EDf being Element of S such that
A5: ( M . (EDf `) = 0 & dom f = EDf & f is_measurable_on EDf ) ;
 ex g1 being PartFunc of X,REAL st
( g = g1 & ex NDg being Element of S st
( M . (NDg `) = 0 & dom g1 = NDg & g1 is_measurable_on NDg & (abs g1) to_power m is_integrable_on M ) ) by A1;
then consider EDg being Element of S such that
A6: ( M . (EDg `) = 0 & dom g = EDg & g is_measurable_on EDg ) ;
set E = EDf /\ EDg;
A7: f + g in Lp_Functions (M,m) by A1, Th25;
then A8: ex h1 being PartFunc of X,REAL st
( f + g = h1 & ex NDfg being Element of S st
( M . (NDfg `) = 0 & dom h1 = NDfg & h1 is_measurable_on NDfg & (abs h1) to_power m is_integrable_on M ) ) ;
A9: dom (f + g) = EDf /\ EDg by A5, A6, VALUED_1:def_1;
then A10: abs (f + g) is_measurable_on EDf /\ EDg by A8, MESFUNC6:48;
reconsider s1 = Integral (M,((abs f) to_power m)) as Real by A1, Th49;
reconsider s2 = Integral (M,((abs g) to_power m)) as Real by A1, Th49;
reconsider s3 = Integral (M,((abs (f + g)) to_power m)) as Real by A7, Th49;
set t = (abs (f + g)) to_power (m - 1);
A11: dom ((abs (f + g)) to_power (m - 1)) = dom (abs (f + g)) by MESFUN6C:def_4;
then A12: dom ((abs (f + g)) to_power (m - 1)) = EDf /\ EDg by A9, VALUED_1:def_11;
then A13: (abs (f + g)) to_power (m - 1) is_measurable_on EDf /\ EDg by A3, A10, A11, MESFUN6C:29;
A14: ((abs (f + g)) to_power (m - 1)) to_power n = (abs (f + g)) to_power m by A2, A3, Th6;
A15: abs ((abs (f + g)) to_power (m - 1)) = (abs (f + g)) to_power (m - 1) by Th14, A4;
then A16: (abs (f + g)) to_power (m - 1) in Lp_Functions (M,n) by A9, A12, A14, A8, A13;
then reconsider s4 = Integral (M,((abs ((abs (f + g)) to_power (m - 1))) to_power n)) as Real by Th49;
 ( ((abs (f + g)) to_power (m - 1)) (#) f is_integrable_on M & ((abs (f + g)) to_power (m - 1)) (#) g is_integrable_on M ) by A1, A16, Th59;
then reconsider u1 = Integral (M,(abs (((abs (f + g)) to_power (m - 1)) (#) f))), u2 = Integral (M,(abs (((abs (f + g)) to_power (m - 1)) (#) g))) as Real by LPSPACE1:44;
A17: ( dom (abs f) = EDf & dom (abs g) = EDg ) by A5, A6, VALUED_1:def_11;
 ( dom (((abs (f + g)) to_power (m - 1)) (#) (abs f)) = (dom ((abs (f + g)) to_power (m - 1))) /\ (dom (abs f)) & dom (((abs (f + g)) to_power (m - 1)) (#) (abs g)) = (dom ((abs (f + g)) to_power (m - 1))) /\ (dom (abs g)) ) by VALUED_1:def_4;
then A18: ( dom (((abs (f + g)) to_power (m - 1)) (#) (abs f)) = EDf /\ EDg & dom (((abs (f + g)) to_power (m - 1)) (#) (abs g)) = EDf /\ EDg ) by A12, A17, XBOOLE_1:17, XBOOLE_1:28;
A19: ( abs (((abs (f + g)) to_power (m - 1)) (#) f) = ((abs (f + g)) to_power (m - 1)) (#) (abs f) & abs (((abs (f + g)) to_power (m - 1)) (#) g) = ((abs (f + g)) to_power (m - 1)) (#) (abs g) & abs (((abs (f + g)) to_power (m - 1)) (#) (f + g)) = ((abs (f + g)) to_power (m - 1)) (#) (abs (f + g)) ) by A15, RFUNCT_1:24;
 ( ((abs (f + g)) to_power (m - 1)) (#) f is_integrable_on M & ((abs (f + g)) to_power (m - 1)) (#) g is_integrable_on M & ((abs (f + g)) to_power (m - 1)) (#) (f + g) is_integrable_on M ) by A1, A16, A7, Th59;
then A20: ( ((abs (f + g)) to_power (m - 1)) (#) (abs f) is_integrable_on M & ((abs (f + g)) to_power (m - 1)) (#) (abs g) is_integrable_on M & ((abs (f + g)) to_power (m - 1)) (#) (abs (f + g)) is_integrable_on M ) by A19, LPSPACE1:44;
set F = ((abs (f + g)) to_power (m - 1)) (#) (abs (f + g));
set G = (((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g));
A21: dom (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) = (EDf /\ EDg) /\ (EDf /\ EDg) by A11, A12, VALUED_1:def_4;
A22: dom ((((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g))) = (EDf /\ EDg) /\ (EDf /\ EDg) by A18, VALUED_1:def_1;
 R_EAL (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) is_integrable_on M by A20, MESFUNC6:def_4;
then  ex E1 being Element of S st
( E1 = dom (R_EAL (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g)))) & R_EAL (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) is_measurable_on E1 ) by MESFUNC5:def_17;
then A23: ((abs (f + g)) to_power (m - 1)) (#) (abs (f + g)) is_measurable_on EDf /\ EDg by A21, MESFUNC6:def_1;
A24: (((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g)) is_integrable_on M by A20, MESFUNC6:100;
 for x being Element of X st x in dom (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) holds
abs ((((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) . x) <= ((((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g))) . x
proof
let x be Element of X; ::_thesis: ( x in dom (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) implies abs ((((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) . x) <= ((((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g))) . x )
assume A25: x in dom (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) ; ::_thesis: abs ((((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) . x) <= ((((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g))) . x
then  abs ((f . x) + (g . x)) = abs ((f + g) . x) by A9, A21, VALUED_1:def_1;
then A26: ( abs ((f . x) + (g . x)) = (abs (f + g)) . x & abs (f . x) = (abs f) . x & abs (g . x) = (abs g) . x ) by VALUED_1:18;
 (R_EAL (f . x)) + (R_EAL (g . x)) = (f . x) + (g . x) by SUPINF_2:1;
then A27: ( |.(R_EAL (f . x)).| = abs (f . x) & |.(R_EAL (g . x)).| = abs (g . x) & |.((R_EAL (f . x)) + (R_EAL (g . x))).| = abs ((f . x) + (g . x)) ) by EXTREAL2:1;
A28: ( ((abs (f + g)) to_power (m - 1)) . x >= 0 & (abs (f + g)) . x >= 0 ) by A3, MESFUNC6:51;
 |.((R_EAL (f . x)) + (R_EAL (g . x))).| <= |.(R_EAL (f . x)).| + |.(R_EAL (g . x)).| by EXTREAL2:13;
then  abs ((f . x) + (g . x)) <= (abs (f . x)) + (abs (g . x)) by A27, SUPINF_2:1;
then A29: (((abs (f + g)) to_power (m - 1)) . x) * ((abs (f + g)) . x) <= (((abs (f + g)) to_power (m - 1)) . x) * (((abs f) . x) + ((abs g) . x)) by A26, A28, XREAL_1:64;
 ( (((abs (f + g)) to_power (m - 1)) . x) * ((abs f) . x) = (((abs (f + g)) to_power (m - 1)) (#) (abs f)) . x & (((abs (f + g)) to_power (m - 1)) . x) * ((abs g) . x) = (((abs (f + g)) to_power (m - 1)) (#) (abs g)) . x ) by VALUED_1:5;
then  (((abs (f + g)) to_power (m - 1)) . x) * (((abs f) . x) + ((abs g) . x)) = ((((abs (f + g)) to_power (m - 1)) (#) (abs f)) . x) + ((((abs (f + g)) to_power (m - 1)) (#) (abs g)) . x) ;
then A30: (((abs (f + g)) to_power (m - 1)) . x) * (((abs f) . x) + ((abs g) . x)) = ((((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g))) . x by A21, A22, A25, VALUED_1:def_1;
 (((abs (f + g)) to_power (m - 1)) . x) * ((abs (f + g)) . x) = (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) . x by VALUED_1:5;
hence  abs ((((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))) . x) <= ((((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g))) . x by A30, A29, A28, ABSVALUE:def_1; ::_thesis: verum
end;
then A31: Integral (M,(abs (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))))) <= Integral (M,((((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g)))) by A21, A22, A23, A24, MESFUNC6:96;
A32: ex E1 being Element of S st
( E1 = (EDf /\ EDg) /\ (EDf /\ EDg) & Integral (M,((((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g)))) = (Integral (M,((((abs (f + g)) to_power (m - 1)) (#) (abs f)) | E1))) + (Integral (M,((((abs (f + g)) to_power (m - 1)) (#) (abs g)) | E1))) ) by A18, A20, MESFUNC6:101;
 ( Integral (M,((((abs (f + g)) to_power (m - 1)) (#) (abs f)) | (EDf /\ EDg))) = Integral (M,(((abs (f + g)) to_power (m - 1)) (#) (abs f))) & Integral (M,((((abs (f + g)) to_power (m - 1)) (#) (abs g)) | (EDf /\ EDg))) = Integral (M,(((abs (f + g)) to_power (m - 1)) (#) (abs g))) ) by A18, RELAT_1:69;
then A33: Integral (M,((((abs (f + g)) to_power (m - 1)) (#) (abs f)) + (((abs (f + g)) to_power (m - 1)) (#) (abs g)))) = u1 + u2 by A19, A32, SUPINF_2:1;
set v1 = (s4 to_power (1 / n)) * (s1 to_power (1 / m));
set v2 = (s4 to_power (1 / n)) * (s2 to_power (1 / m));
 ( ex r4 being Real st
( r4 = Integral (M,((abs ((abs (f + g)) to_power (m - 1))) to_power n)) & ex r1 being Real st
( r1 = Integral (M,((abs f) to_power m)) & Integral (M,(abs (((abs (f + g)) to_power (m - 1)) (#) f))) <= (r4 to_power (1 / n)) * (r1 to_power (1 / m)) ) ) & ex r4 being Real st
( r4 = Integral (M,((abs ((abs (f + g)) to_power (m - 1))) to_power n)) & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power m)) & Integral (M,(abs (((abs (f + g)) to_power (m - 1)) (#) g))) <= (r4 to_power (1 / n)) * (r2 to_power (1 / m)) ) ) ) by A1, A16, Th60;
then A34: u1 + u2 <= ((s4 to_power (1 / n)) * (s1 to_power (1 / m))) + ((s4 to_power (1 / n)) * (s2 to_power (1 / m))) by XREAL_1:7;
((abs (f + g)) to_power (m - 1)) (#) (abs (f + g)) = ((abs (f + g)) to_power (m - 1)) (#) ((abs (f + g)) to_power 1) by Th8
.= (abs (f + g)) to_power ((m - 1) + 1) by Th7, A3 ;
then  Integral (M,(abs (((abs (f + g)) to_power (m - 1)) (#) (abs (f + g))))) = s3 by Th14;
then A35: s3 <= ((s3 to_power (1 / n)) * (s1 to_power (1 / m))) + ((s3 to_power (1 / n)) * (s2 to_power (1 / m))) by A14, A15, A31, A33, A34, XXREAL_0:2;
percases ( s3 = 0 or s3 > 0 ) by A7, Th49;
suppose s3 = 0 ; ::_thesis: ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) )
then A36: s3 to_power (1 / m) = 0 by POWER:def_2;
 ( s1 to_power (1 / m) >= 0 & s2 to_power (1 / m) >= 0 ) by A1, Th49, Th4;
then  s3 to_power (1 / m) <= (s1 to_power (1 / m)) + (s2 to_power (1 / m)) by A36;
hence  ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) ) ; ::_thesis: verum
end;
supposeA37: s3 > 0 ; ::_thesis: ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) )
then A38: s3 to_power (1 / n) > 0 by POWER:34;
set w1 = s3 to_power (1 / n);
 (1 / (s3 to_power (1 / n))) * (((s3 to_power (1 / n)) * (s1 to_power (1 / m))) + ((s3 to_power (1 / n)) * (s2 to_power (1 / m)))) = ((1 / (s3 to_power (1 / n))) * (s3 to_power (1 / n))) * ((s1 to_power (1 / m)) + (s2 to_power (1 / m))) ;
then A39: (1 / (s3 to_power (1 / n))) * (((s3 to_power (1 / n)) * (s1 to_power (1 / m))) + ((s3 to_power (1 / n)) * (s2 to_power (1 / m)))) = 1 * ((s1 to_power (1 / m)) + (s2 to_power (1 / m))) by A38, XCMPLX_1:106;
(1 / (s3 to_power (1 / n))) * s3 = (s3 to_power (- (1 / n))) * s3 by A37, POWER:28
.= (s3 to_power (- (1 / n))) * (s3 to_power 1) by POWER:25
.= s3 to_power ((- (1 / n)) + 1) by A37, POWER:27
.= s3 to_power (1 / m) by A1 ;
hence  ex r1, r2, r3 being Real st
( r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) & r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) ) by A38, A39, A35, XREAL_1:64; ::_thesis: verum
end;
end;
end;

theorem Th61: :: LPSPACE2:61
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m being positive Real
for r1, r2, r3 being Element of REAL st 1 <= m & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) & r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) holds
r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m))
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m being positive Real
for r1, r2, r3 being Element of REAL st 1 <= m & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) & r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) holds
r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m))
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for m being positive Real
for r1, r2, r3 being Element of REAL st 1 <= m & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) & r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) holds
r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m))
let M be sigma_Measure of S; ::_thesis: for f, g being PartFunc of X,REAL
for m being positive Real
for r1, r2, r3 being Element of REAL st 1 <= m & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) & r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) holds
r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m))
let f, g be PartFunc of X,REAL; ::_thesis: for m being positive Real
for r1, r2, r3 being Element of REAL st 1 <= m & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) & r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) holds
r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m))
let m be positive Real; ::_thesis: for r1, r2, r3 being Element of REAL st 1 <= m & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) & r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) holds
r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m))
let r1, r2, r3 be Element of REAL ; ::_thesis: ( 1 <= m & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) & r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) implies r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) )
assume A1: ( 1 <= m & f in Lp_Functions (M,m) & g in Lp_Functions (M,m) & r1 = Integral (M,((abs f) to_power m)) & r2 = Integral (M,((abs g) to_power m)) & r3 = Integral (M,((abs (f + g)) to_power m)) ) ; ::_thesis: r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m))
percases ( m = 1 or m <> 1 ) ;
supposeA2: m = 1 ; ::_thesis: r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m))
then A3: ( r1 = Integral (M,(abs f)) & r2 = Integral (M,(abs g)) & r3 = Integral (M,(abs (f + g))) ) by A1, Th8;
A4: ex f1 being PartFunc of X,REAL st
( f = f1 & ex ND being Element of S st
( M . (ND `) = 0 & dom f1 = ND & f1 is_measurable_on ND & (abs f1) to_power m is_integrable_on M ) ) by A1;
A5: ex g1 being PartFunc of X,REAL st
( g = g1 & ex ND being Element of S st
( M . (ND `) = 0 & dom g1 = ND & g1 is_measurable_on ND & (abs g1) to_power m is_integrable_on M ) ) by A1;
then  ( abs f is_integrable_on M & abs g is_integrable_on M ) by A2, A4, Th8;
then  ( f is_integrable_on M & g is_integrable_on M ) by A4, A5, MESFUNC6:94;
then  Integral (M,(abs (f + g))) <= (Integral (M,(abs f))) + (Integral (M,(abs g))) by LPSPACE1:55;
then A6: r3 <= r1 + r2 by A3, XXREAL_3:def_2;
 ( r1 to_power (1 / m) = r1 & r2 to_power (1 / m) = r2 ) by A2, POWER:25;
hence  r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) by A6, A2, POWER:25; ::_thesis: verum
end;
supposeA7: m <> 1 ; ::_thesis: r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m))
set n1 = 1 - (1 / m);
 1 < m by A1, A7, XXREAL_0:1;
then  1 / m < 1 by XREAL_1:189;
then  0 < 1 - (1 / m) by XREAL_1:50;
then reconsider n = 1 / (1 - (1 / m)) as positive Real ;
 (1 / m) + (1 / n) = 1 ;
then  ex rr1, rr2, rr3 being Real st
( rr1 = Integral (M,((abs f) to_power m)) & rr2 = Integral (M,((abs g) to_power m)) & rr3 = Integral (M,((abs (f + g)) to_power m)) & rr3 to_power (1 / m) <= (rr1 to_power (1 / m)) + (rr2 to_power (1 / m)) ) by A1, Lm5;
hence  r3 to_power (1 / m) <= (r1 to_power (1 / m)) + (r2 to_power (1 / m)) by A1; ::_thesis: verum
end;
end;
end;

Lm6:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being geq_than_1 Real holds
( Lp-Space (M,k) is reflexive & Lp-Space (M,k) is discerning & Lp-Space (M,k) is RealNormSpace-like & Lp-Space (M,k) is vector-distributive & Lp-Space (M,k) is scalar-distributive & Lp-Space (M,k) is scalar-associative & Lp-Space (M,k) is scalar-unital & Lp-Space (M,k) is Abelian & Lp-Space (M,k) is add-associative & Lp-Space (M,k) is right_zeroed & Lp-Space (M,k) is right_complementable )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being geq_than_1 Real holds
( Lp-Space (M,k) is reflexive & Lp-Space (M,k) is discerning & Lp-Space (M,k) is RealNormSpace-like & Lp-Space (M,k) is vector-distributive & Lp-Space (M,k) is scalar-distributive & Lp-Space (M,k) is scalar-associative & Lp-Space (M,k) is scalar-unital & Lp-Space (M,k) is Abelian & Lp-Space (M,k) is add-associative & Lp-Space (M,k) is right_zeroed & Lp-Space (M,k) is right_complementable )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being geq_than_1 Real holds
( Lp-Space (M,k) is reflexive & Lp-Space (M,k) is discerning & Lp-Space (M,k) is RealNormSpace-like & Lp-Space (M,k) is vector-distributive & Lp-Space (M,k) is scalar-distributive & Lp-Space (M,k) is scalar-associative & Lp-Space (M,k) is scalar-unital & Lp-Space (M,k) is Abelian & Lp-Space (M,k) is add-associative & Lp-Space (M,k) is right_zeroed & Lp-Space (M,k) is right_complementable )
let M be sigma_Measure of S; ::_thesis: for k being geq_than_1 Real holds
( Lp-Space (M,k) is reflexive & Lp-Space (M,k) is discerning & Lp-Space (M,k) is RealNormSpace-like & Lp-Space (M,k) is vector-distributive & Lp-Space (M,k) is scalar-distributive & Lp-Space (M,k) is scalar-associative & Lp-Space (M,k) is scalar-unital & Lp-Space (M,k) is Abelian & Lp-Space (M,k) is add-associative & Lp-Space (M,k) is right_zeroed & Lp-Space (M,k) is right_complementable )
let k be geq_than_1 Real; ::_thesis: ( Lp-Space (M,k) is reflexive & Lp-Space (M,k) is discerning & Lp-Space (M,k) is RealNormSpace-like & Lp-Space (M,k) is vector-distributive & Lp-Space (M,k) is scalar-distributive & Lp-Space (M,k) is scalar-associative & Lp-Space (M,k) is scalar-unital & Lp-Space (M,k) is Abelian & Lp-Space (M,k) is add-associative & Lp-Space (M,k) is right_zeroed & Lp-Space (M,k) is right_complementable )
set x = 0. (Lp-Space (M,k));
 0. (Lp-Space (M,k)) = 0. (Pre-Lp-Space (M,k)) ;
then  0. (Lp-Space (M,k)) = zeroCoset (M,k) by Def11;
then  X --> 0 in 0. (Lp-Space (M,k)) by Th38, Th23;
then  ex r being Real st
( 0 <= r & r = Integral (M,((abs (X --> 0)) to_power k)) & ||.(0. (Lp-Space (M,k))).|| = r to_power (1 / k) ) by Th55;
then consider r0 being Real such that
A1: ( r0 = Integral (M,((abs (X --> 0)) to_power k)) & (Lp-Norm (M,k)) . (0. (Lp-Space (M,k))) = r0 to_power (1 / k) ) ;
 r0 = 0 by A1, Th58;
hence  ||.(0. (Lp-Space (M,k))).|| = 0 by A1, POWER:def_2; :: according to NORMSP_0:def_6 ::_thesis: ( Lp-Space (M,k) is discerning & Lp-Space (M,k) is RealNormSpace-like & Lp-Space (M,k) is vector-distributive & Lp-Space (M,k) is scalar-distributive & Lp-Space (M,k) is scalar-associative & Lp-Space (M,k) is scalar-unital & Lp-Space (M,k) is Abelian & Lp-Space (M,k) is add-associative & Lp-Space (M,k) is right_zeroed & Lp-Space (M,k) is right_complementable )
now__::_thesis:_for_x,_y_being_Point_of_(Lp-Space_(M,k))
for_a_being_Real_holds_
(_(_||.x.||_=_0_implies_x_=_0._(Lp-Space_(M,k))_)_&_0_<=_||.x.||_&_||.(x_+_y).||_<=_||.x.||_+_||.y.||_&_||.(a_*_x).||_=_(abs_a)_*_||.x.||_)
let x, y be Point of (Lp-Space (M,k)); ::_thesis: for a being Real holds
( ( ||.x.|| = 0 implies x = 0. (Lp-Space (M,k)) ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = (abs a) * ||.x.|| )
let a be Real; ::_thesis: ( ( ||.x.|| = 0 implies x = 0. (Lp-Space (M,k)) ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = (abs a) * ||.x.|| )
A2: 1 <= k by Def1;
hereby  ::_thesis: ( 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = (abs a) * ||.x.|| )
assume A3: ||.x.|| = 0 ; ::_thesis: x = 0. (Lp-Space (M,k))
consider f being PartFunc of X,REAL such that
A4: ( f in x & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & ||.x.|| = r to_power (1 / k) ) ) by Def12;
A5: f in Lp_Functions (M,k) by Th51, A4;
then consider r1 being Real such that
A6: ( r1 = Integral (M,((abs f) to_power k)) & r1 >= 0 & (Lp-Norm (M,k)) . x = r1 to_power (1 / k) ) by A4, Th49;
 r1 = 0 by A3, A6, POWER:34;
then  zeroCoset (M,k) = a.e-eq-class_Lp (f,M,k) by A5, A6, Th57, Th42;
then  0. (Pre-Lp-Space (M,k)) = a.e-eq-class_Lp (f,M,k) by Def11;
hence  x = 0. (Lp-Space (M,k)) by A4, Th55; ::_thesis: verum
end;
consider f being PartFunc of X,REAL such that
A7: ( f in x & ex r1 being Real st
( r1 = Integral (M,((abs f) to_power k)) & ||.x.|| = r1 to_power (1 / k) ) ) by Def12;
A8: ( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) ) by Th51, A7;
consider g being PartFunc of X,REAL such that
A9: ( g in y & ex r2 being Real st
( r2 = Integral (M,((abs g) to_power k)) & ||.y.|| = r2 to_power (1 / k) ) ) by Def12;
A10: ( (abs g) to_power k is_integrable_on M & g in Lp_Functions (M,k) ) by Th51, A9;
consider s1 being Real such that
A11: ( s1 = Integral (M,((abs f) to_power k)) & ||.x.|| = s1 to_power (1 / k) ) by A7;
A12: ( s1 = 0 implies s1 to_power (1 / k) >= 0 ) by POWER:def_2;
 ( s1 > 0 implies s1 to_power (1 / k) >= 0 ) by POWER:34;
hence  0 <= ||.x.|| by A12, A8, A11, Th49; ::_thesis: ( ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = (abs a) * ||.x.|| )
set t = f + g;
set w = x + y;
A13: s1 >= 0 by A8, A11, Th49;
consider s2 being Real such that
A14: ( s2 = Integral (M,((abs g) to_power k)) & ||.y.|| = s2 to_power (1 / k) ) by A9;
 f + g in x + y by Th54, A7, A9;
then  ex r being Real st
( 0 <= r & r = Integral (M,((abs (f + g)) to_power k)) & ||.(x + y).|| = r to_power (1 / k) ) by Th53;
hence  ||.(x + y).|| <= ||.x.|| + ||.y.|| by Th61, A2, A8, A10, A14, A11; ::_thesis: ||.(a * x).|| = (abs a) * ||.x.||
set t = a (#) f;
set w = a * x;
 a (#) f in a * x by Th54, A7;
then  ex r being Real st
( 0 <= r & r = Integral (M,((abs (a (#) f)) to_power k)) & ||.(a * x).|| = r to_power (1 / k) ) by Th53;
then consider s being Real such that
A15: ( s = Integral (M,((abs (a (#) f)) to_power k)) & ||.(a * x).|| = s to_power (1 / k) ) ;
A16: s = Integral (M,(((abs a) to_power k) (#) ((abs f) to_power k))) by A15, Th18
.= (R_EAL ((abs a) to_power k)) * (R_EAL s1) by A11, A8, MESFUNC6:102
.= ((abs a) to_power k) * s1 by EXTREAL1:5 ;
 (abs a) to_power k >= 0 by Th4, COMPLEX1:46;
then ||.(a * x).|| = (((abs a) to_power k) to_power (1 / k)) * (s1 to_power (1 / k)) by A13, A15, A16, Th5
.= ((abs a) to_power (k * (1 / k))) * (s1 to_power (1 / k)) by COMPLEX1:46, HOLDER_1:2
.= ((abs a) to_power 1) * (s1 to_power (1 / k)) by XCMPLX_1:106 ;
hence  ||.(a * x).|| = (abs a) * ||.x.|| by A11, POWER:25; ::_thesis: verum
end;
hence  ( Lp-Space (M,k) is discerning & Lp-Space (M,k) is RealNormSpace-like & Lp-Space (M,k) is vector-distributive & Lp-Space (M,k) is scalar-distributive & Lp-Space (M,k) is scalar-associative & Lp-Space (M,k) is scalar-unital & Lp-Space (M,k) is Abelian & Lp-Space (M,k) is add-associative & Lp-Space (M,k) is right_zeroed & Lp-Space (M,k) is right_complementable ) by NORMSP_0:def_5, NORMSP_1:def_1, RSSPACE3:2; ::_thesis: verum
end;

registration
let k be geq_than_1 Real;
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
cluster Lp-Space (M,k) ->  non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ;
coherence
 ( Lp-Space (M,k) is reflexive & Lp-Space (M,k) is discerning & Lp-Space (M,k) is RealNormSpace-like & Lp-Space (M,k) is vector-distributive & Lp-Space (M,k) is scalar-distributive & Lp-Space (M,k) is scalar-associative & Lp-Space (M,k) is scalar-unital & Lp-Space (M,k) is Abelian & Lp-Space (M,k) is add-associative & Lp-Space (M,k) is right_zeroed & Lp-Space (M,k) is right_complementable ) by Lm6;
end;

begin

theorem Th62: :: LPSPACE2:62
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for Sq being sequence of (Lp-Space (M,k)) ex Fsq being Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for Sq being sequence of (Lp-Space (M,k)) ex Fsq being Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real
for Sq being sequence of (Lp-Space (M,k)) ex Fsq being Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
let M be sigma_Measure of S; ::_thesis: for k being positive Real
for Sq being sequence of (Lp-Space (M,k)) ex Fsq being Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
let k be positive Real; ::_thesis: for Sq being sequence of (Lp-Space (M,k)) ex Fsq being Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
let Sq be sequence of (Lp-Space (M,k)); ::_thesis: ex Fsq being Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
defpred S1[ Element of NAT , set ] means  ex f being PartFunc of X,REAL st
( $2 = f & f in Lp_Functions (M,k) & f in Sq . $1 & Sq . $1 = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & ||.(Sq . $1).|| = r to_power (1 / k) ) );
A1: for x being Element of NAT ex y being Element of PFuncs (X,REAL) st S1[x,y]
proof
let x be Element of NAT ; ::_thesis: ex y being Element of PFuncs (X,REAL) st S1[x,y]
consider y being PartFunc of X,REAL such that
A2: ( y in Lp_Functions (M,k) & Sq . x = a.e-eq-class_Lp (y,M,k) ) by Th53;
 ex r being Real st
( 0 <= r & r = Integral (M,((abs y) to_power k)) & ||.(Sq . x).|| = r to_power (1 / k) ) by Th53, A2, Th38;
hence  ex y being Element of PFuncs (X,REAL) st S1[x,y] by A2, Th38; ::_thesis: verum
end;
consider G being Function of NAT,(PFuncs (X,REAL)) such that
A3: for n being Element of NAT holds S1[n,G . n] from FUNCT_2:sch_3(A1);
reconsider G = G as Functional_Sequence of X,REAL ;
now__::_thesis:_for_n_being_Element_of_NAT_holds_
(_G_._n_in_Lp_Functions_(M,k)_&_G_._n_in_Sq_._n_&_Sq_._n_=_a.e-eq-class_Lp_((G_._n),M,k)_&_ex_r_being_Real_st_
(_r_=_Integral_(M,((abs_(G_._n))_to_power_k))_&_||.(Sq_._n).||_=_r_to_power_(1_/_k)_)_)
let n be Element of NAT ; ::_thesis: ( G . n in Lp_Functions (M,k) & G . n in Sq . n & Sq . n = a.e-eq-class_Lp ((G . n),M,k) & ex r being Real st
( r = Integral (M,((abs (G . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
 ex f being PartFunc of X,REAL st
( G . n = f & f in Lp_Functions (M,k) & f in Sq . n & Sq . n = a.e-eq-class_Lp (f,M,k) & ex r being Real st
( r = Integral (M,((abs f) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) ) by A3;
hence  ( G . n in Lp_Functions (M,k) & G . n in Sq . n & Sq . n = a.e-eq-class_Lp ((G . n),M,k) & ex r being Real st
( r = Integral (M,((abs (G . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) ) ; ::_thesis: verum
end;
hence  ex Fsq being Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) ) ; ::_thesis: verum
end;

theorem Th63: :: LPSPACE2:63
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for Sq being sequence of (Lp-Space (M,k)) ex Fsq being with_the_same_dom Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for Sq being sequence of (Lp-Space (M,k)) ex Fsq being with_the_same_dom Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real
for Sq being sequence of (Lp-Space (M,k)) ex Fsq being with_the_same_dom Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
let M be sigma_Measure of S; ::_thesis: for k being positive Real
for Sq being sequence of (Lp-Space (M,k)) ex Fsq being with_the_same_dom Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
let k be positive Real; ::_thesis: for Sq being sequence of (Lp-Space (M,k)) ex Fsq being with_the_same_dom Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
let Sq be sequence of (Lp-Space (M,k)); ::_thesis: ex Fsq being with_the_same_dom Functional_Sequence of X,REAL st
for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
consider Fsq being Functional_Sequence of X,REAL such that
A1: for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) ) by Th62;
defpred S1[ Element of NAT , set ] means  ex DMFSQN being Element of S st
( $2 = DMFSQN & ex FSQN being PartFunc of X,REAL st
( Fsq . $1 = FSQN & M . (DMFSQN `) = 0 & dom FSQN = DMFSQN & FSQN is_measurable_on DMFSQN & (abs FSQN) to_power k is_integrable_on M ) );
A2: for n being Element of NAT ex y being Element of S st S1[n,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of S st S1[n,y]
 Fsq . n in Lp_Functions (M,k) by A1;
then  ex FMF being PartFunc of X,REAL st
( Fsq . n = FMF & ex ND being Element of S st
( M . (ND `) = 0 & dom FMF = ND & FMF is_measurable_on ND & (abs FMF) to_power k is_integrable_on M ) ) ;
hence  ex y being Element of S st S1[n,y] ; ::_thesis: verum
end;
consider G being Function of NAT,S such that
A3: for n being Element of NAT holds S1[n,G . n] from FUNCT_2:sch_3(A2);
reconsider E0 = meet (rng G) as Element of S ;
A4: for n being Element of NAT holds
( M . (X \ (G . n)) = 0 & E0 c= dom (Fsq . n) )
proof
let n be Element of NAT ; ::_thesis: ( M . (X \ (G . n)) = 0 & E0 c= dom (Fsq . n) )
 ex D being Element of S st
( G . n = D & ex F being PartFunc of X,REAL st
( Fsq . n = F & M . (D `) = 0 & dom F = D & F is_measurable_on D & (abs F) to_power k is_integrable_on M ) ) by A3;
hence  ( M . (X \ (G . n)) = 0 & E0 c= dom (Fsq . n) ) by FUNCT_2:4, SETFAM_1:3; ::_thesis: verum
end;
A5: X \ (rng G) is N_Sub_set_fam of X by MEASURE1:21;
 for A being set st A in X \ (rng G) holds
( A in S & A is measure_zero of M )
proof
let A be set ; ::_thesis: ( A in X \ (rng G) implies ( A in S & A is measure_zero of M ) )
assume A6: A in X \ (rng G) ; ::_thesis: ( A in S & A is measure_zero of M )
then reconsider A0 = A as Subset of X ;
 A0 ` in rng G by A6, SETFAM_1:def_7;
then consider n being set  such that
A7: ( n in NAT & A0 ` = G . n ) by FUNCT_2:11;
reconsider n = n as Element of NAT by A7;
A8: (A0 `) ` = A0 ;
then  A0 = X \ (G . n) by A7;
hence  A in S by MEASURE1:34; ::_thesis: A is measure_zero of M
A9: M . A0 = 0 by A4, A7, A8;
 A0 = X \ (G . n) by A7, A8;
then  A is Element of S by MEASURE1:34;
hence  A is measure_zero of M by A9, MEASURE1:def_7; ::_thesis: verum
end;
then A10: ( ( for A being set st A in X \ (rng G) holds
A in S ) & ( for A being set st A in X \ (rng G) holds
A is measure_zero of M ) ) ;
then  X \ (rng G) c= S by TARSKI:def_3;
then  X \ (rng G) is N_Measure_fam of S by A5, MEASURE2:def_1;
then A11: union (X \ (rng G)) is measure_zero of M by A10, MEASURE2:14;
E0 ` = X \ (X \ (union (X \ (rng G)))) by MEASURE1:4
.= X /\ (union (X \ (rng G))) by XBOOLE_1:48
.= union (X \ (rng G)) by XBOOLE_1:28 ;
then A12: M . (E0 `) = 0 by A11, MEASURE1:def_7;
set Fsq2 = Fsq || E0;
A13: for n being Element of NAT holds dom ((Fsq || E0) . n) = E0
proof
let n be Element of NAT ; ::_thesis: dom ((Fsq || E0) . n) = E0
 dom ((Fsq || E0) . n) = dom ((Fsq . n) | E0) by MESFUN9C:def_1;
then  dom ((Fsq || E0) . n) = (dom (Fsq . n)) /\ E0 by RELAT_1:61;
hence  dom ((Fsq || E0) . n) = E0 by A4, XBOOLE_1:28; ::_thesis: verum
end;
now__::_thesis:_for_n,_m_being_Nat_holds_dom_((Fsq_||_E0)_._n)_=_dom_((Fsq_||_E0)_._m)
let n, m be Nat; ::_thesis: dom ((Fsq || E0) . n) = dom ((Fsq || E0) . m)
 ( n is Element of NAT & m is Element of NAT ) by ORDINAL1:def_12;
then  ( dom ((Fsq || E0) . n) = E0 & dom ((Fsq || E0) . m) = E0 ) by A13;
hence  dom ((Fsq || E0) . n) = dom ((Fsq || E0) . m) ; ::_thesis: verum
end;
then reconsider Fsq2 = Fsq || E0 as with_the_same_dom Functional_Sequence of X,REAL by MESFUNC8:def_2;
take  Fsq2 ; ::_thesis: for n being Element of NAT holds
( Fsq2 . n in Lp_Functions (M,k) & Fsq2 . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq2 . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq2 . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
hereby  ::_thesis: verum
let n be Element of NAT ; ::_thesis: ( Fsq2 . n in Lp_Functions (M,k) & Fsq2 . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq2 . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq2 . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
 Fsq . n in Lp_Functions (M,k) by A1;
then A14: ex FMF being PartFunc of X,REAL st
( Fsq . n = FMF & ex ND being Element of S st
( M . (ND `) = 0 & dom FMF = ND & FMF is_measurable_on ND & (abs FMF) to_power k is_integrable_on M ) ) ;
then reconsider E2 = dom (Fsq . n) as Element of S ;
A15: E2 /\ E0 = E0 by A4, XBOOLE_1:28;
 R_EAL (Fsq . n) is_measurable_on E2 by A14, MESFUNC6:def_1;
then  R_EAL (Fsq . n) is_measurable_on E0 by A4, MESFUNC1:30;
then  Fsq . n is_measurable_on E0 by MESFUNC6:def_1;
then  (Fsq . n) | E0 is_measurable_on E0 by A15, MESFUNC6:76;
then A16: Fsq2 . n is_measurable_on E0 by MESFUN9C:def_1;
A17: dom (Fsq2 . n) = E0 by A13;
 ( dom ((abs (Fsq . n)) to_power k) = dom (abs (Fsq . n)) & dom ((abs (Fsq2 . n)) to_power k) = dom (abs (Fsq2 . n)) ) by MESFUN6C:def_4;
then A18: ( dom ((abs (Fsq . n)) to_power k) = dom (Fsq . n) & dom ((abs (Fsq2 . n)) to_power k) = dom (Fsq2 . n) ) by VALUED_1:def_11;
 for x being set st x in dom ((abs (Fsq2 . n)) to_power k) holds
((abs (Fsq2 . n)) to_power k) . x = ((abs (Fsq . n)) to_power k) . x
proof
let x be set ; ::_thesis: ( x in dom ((abs (Fsq2 . n)) to_power k) implies ((abs (Fsq2 . n)) to_power k) . x = ((abs (Fsq . n)) to_power k) . x )
assume A19: x in dom ((abs (Fsq2 . n)) to_power k) ; ::_thesis: ((abs (Fsq2 . n)) to_power k) . x = ((abs (Fsq . n)) to_power k) . x
then reconsider x0 = x as Element of X ;
A20: x in dom ((abs (Fsq . n)) to_power k) by A17, A18, A15, A19, XBOOLE_0:def_4;
thus ((abs (Fsq2 . n)) to_power k) . x = ((abs (Fsq2 . n)) . x0) to_power k by A19, MESFUN6C:def_4
.= (abs ((Fsq2 . n) . x0)) to_power k by VALUED_1:18
.= (abs (((Fsq . n) | E0) . x0)) to_power k by MESFUN9C:def_1
.= (abs ((Fsq . n) . x0)) to_power k by A17, A18, A19, FUNCT_1:49
.= ((abs (Fsq . n)) . x0) to_power k by VALUED_1:18
.= ((abs (Fsq . n)) to_power k) . x by A20, MESFUN6C:def_4 ; ::_thesis: verum
end;
then  ((abs (Fsq . n)) to_power k) | E0 = (abs (Fsq2 . n)) to_power k by A13, A15, A18, FUNCT_1:46;
then  (abs (Fsq2 . n)) to_power k is_integrable_on M by A14, MESFUNC6:91;
hence A21: Fsq2 . n in Lp_Functions (M,k) by A16, A17, A12; ::_thesis: ( Fsq2 . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq2 . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq2 . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
A22: ( Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) ) by A1;
reconsider EB = E0 ` as Element of S by MEASURE1:34;
 (Fsq2 . n) | (EB `) = Fsq2 . n by A17, RELAT_1:68;
then  (Fsq2 . n) | (EB `) = (Fsq . n) | (EB `) by MESFUN9C:def_1;
then A23: Fsq2 . n a.e.= Fsq . n,M by A12, LPSPACE1:def_10;
hence  Fsq2 . n in Sq . n by A22, A21, Th36; ::_thesis: ( Sq . n = a.e-eq-class_Lp ((Fsq2 . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq2 . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) )
 a.e-eq-class_Lp ((Fsq2 . n),M,k) = a.e-eq-class_Lp ((Fsq . n),M,k) by Th42, A23;
hence  Sq . n = a.e-eq-class_Lp ((Fsq2 . n),M,k) by A1; ::_thesis: ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq2 . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) )
hence  ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq2 . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) by Th53, Th38, A21; ::_thesis: verum
end;
end;

Lm7:  for X being RealNormSpace
for Sq being sequence of X
for Sq0 being Point of X
for R1 being Real_Sequence
for N being V167() sequence of NAT st Sq is Cauchy_sequence_by_Norm & ( for i being Nat holds R1 . i = ||.(Sq0 - (Sq . (N . i))).|| ) & R1 is convergent & lim R1 = 0 holds
( Sq is convergent & lim Sq = Sq0 & ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 )
proof
let X be RealNormSpace; ::_thesis: for Sq being sequence of X
for Sq0 being Point of X
for R1 being Real_Sequence
for N being V167() sequence of NAT st Sq is Cauchy_sequence_by_Norm & ( for i being Nat holds R1 . i = ||.(Sq0 - (Sq . (N . i))).|| ) & R1 is convergent & lim R1 = 0 holds
( Sq is convergent & lim Sq = Sq0 & ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 )
let Sq be sequence of X; ::_thesis: for Sq0 being Point of X
for R1 being Real_Sequence
for N being V167() sequence of NAT st Sq is Cauchy_sequence_by_Norm & ( for i being Nat holds R1 . i = ||.(Sq0 - (Sq . (N . i))).|| ) & R1 is convergent & lim R1 = 0 holds
( Sq is convergent & lim Sq = Sq0 & ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 )
let Sq0 be Point of X; ::_thesis: for R1 being Real_Sequence
for N being V167() sequence of NAT st Sq is Cauchy_sequence_by_Norm & ( for i being Nat holds R1 . i = ||.(Sq0 - (Sq . (N . i))).|| ) & R1 is convergent & lim R1 = 0 holds
( Sq is convergent & lim Sq = Sq0 & ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 )
let R1 be Real_Sequence; ::_thesis: for N being V167() sequence of NAT st Sq is Cauchy_sequence_by_Norm & ( for i being Nat holds R1 . i = ||.(Sq0 - (Sq . (N . i))).|| ) & R1 is convergent & lim R1 = 0 holds
( Sq is convergent & lim Sq = Sq0 & ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 )
let N be V167() sequence of NAT; ::_thesis: ( Sq is Cauchy_sequence_by_Norm & ( for i being Nat holds R1 . i = ||.(Sq0 - (Sq . (N . i))).|| ) & R1 is convergent & lim R1 = 0 implies ( Sq is convergent & lim Sq = Sq0 & ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 ) )
assume that
A1: Sq is Cauchy_sequence_by_Norm and
A2: for i being Nat holds R1 . i = ||.(Sq0 - (Sq . (N . i))).|| and
A3: ( R1 is convergent & lim R1 = 0 ) ; ::_thesis: ( Sq is convergent & lim Sq = Sq0 & ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 )
A4: now__::_thesis:_for_p_being_Real_st_0_<_p_holds_
ex_n3_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n3_<=_n_holds_
||.((Sq_._n)_-_Sq0).||_<_p
let p be Real; ::_thesis: ( 0 < p implies ex n3 being Element of NAT st
for n being Element of NAT st n3 <= n holds
||.((Sq . n) - Sq0).|| < p )
assume A5: 0 < p ; ::_thesis: ex n3 being Element of NAT st
for n being Element of NAT st n3 <= n holds
||.((Sq . n) - Sq0).|| < p
then consider n2 being Element of NAT  such that
A6: for m, n being Element of NAT st n2 <= m & n2 <= n holds
||.((Sq . m) - (Sq . n)).|| < p / 2 by A1, RSSPACE3:8;
consider n1 being Element of NAT  such that
A7: for l being Element of NAT st n1 <= l holds
abs ((R1 . l) - 0) < p / 2 by A3, A5, SEQ_2:def_7;
reconsider n3 = max (n1,n2) as Element of NAT ;
take n3 = n3; ::_thesis: for n being Element of NAT st n3 <= n holds
||.((Sq . n) - Sq0).|| < p
thus  for n being Element of NAT st n3 <= n holds
||.((Sq . n) - Sq0).|| < p ::_thesis: verum
proof
let n be Element of NAT ; ::_thesis: ( n3 <= n implies ||.((Sq . n) - Sq0).|| < p )
assume A8: n3 <= n ; ::_thesis: ||.((Sq . n) - Sq0).|| < p
 n1 <= n3 by XXREAL_0:25;
then  n1 <= n by A8, XXREAL_0:2;
then  abs ((R1 . n) - 0) < p / 2 by A7;
then A9: abs ||.(Sq0 - (Sq . (N . n))).|| < p / 2 by A2;
A10: ||.(Sq0 - (Sq . (N . n))).|| < p / 2 by A9, ABSVALUE:def_1;
 n <= N . n by SEQM_3:14;
then A11: n3 <= N . n by A8, XXREAL_0:2;
 n2 <= n3 by XXREAL_0:25;
then  ( n2 <= N . n & n2 <= n ) by A8, A11, XXREAL_0:2;
then  ||.((Sq . (N . n)) - (Sq . n)).|| < p / 2 by A6;
then A12: ||.(Sq0 - (Sq . (N . n))).|| + ||.((Sq . (N . n)) - (Sq . n)).|| < (p / 2) + (p / 2) by A10, XREAL_1:8;
A13: ||.((Sq . n) - Sq0).|| = ||.(Sq0 - (Sq . n)).|| by NORMSP_1:7
.= ||.((Sq0 - (Sq . (N . n))) + ((Sq . (N . n)) - (Sq . n))).|| by LOPBAN_3:3 ;
 ||.((Sq0 - (Sq . (N . n))) + ((Sq . (N . n)) - (Sq . n))).|| <= ||.(Sq0 - (Sq . (N . n))).|| + ||.((Sq . (N . n)) - (Sq . n)).|| by NORMSP_1:def_1;
hence  ||.((Sq . n) - Sq0).|| < p by A13, A12, XXREAL_0:2; ::_thesis: verum
end;
end;
hence A14: Sq is convergent by NORMSP_1:def_6; ::_thesis: ( lim Sq = Sq0 & ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 )
hence  lim Sq = Sq0 by A4, NORMSP_1:def_7; ::_thesis: ( ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 )
hence  ( ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 ) by A14, NORMSP_1:24; ::_thesis: verum
end;

theorem :: LPSPACE2:64
for X being RealNormSpace
for Sq being sequence of X
for Sq0 being Point of X st ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 holds
( Sq is convergent & lim Sq = Sq0 )
proof
let X be RealNormSpace; ::_thesis: for Sq being sequence of X
for Sq0 being Point of X st ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 holds
( Sq is convergent & lim Sq = Sq0 )
let Sq be sequence of X; ::_thesis: for Sq0 being Point of X st ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 holds
( Sq is convergent & lim Sq = Sq0 )
let Sq0 be Point of X; ::_thesis: ( ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 implies ( Sq is convergent & lim Sq = Sq0 ) )
assume A1: ( ||.(Sq - Sq0).|| is convergent & lim ||.(Sq - Sq0).|| = 0 ) ; ::_thesis: ( Sq is convergent & lim Sq = Sq0 )
A2: for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((Sq . m) - Sq0).|| < p
proof
let p be Real; ::_thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((Sq . m) - Sq0).|| < p )
assume  0 < p ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((Sq . m) - Sq0).|| < p
then consider n being Element of NAT  such that
A3: for m being Element of NAT st n <= m holds
abs ((||.(Sq - Sq0).|| . m) - 0) < p by A1, SEQ_2:def_7;
take  n ; ::_thesis: for m being Element of NAT st n <= m holds
||.((Sq . m) - Sq0).|| < p
hereby  ::_thesis: verum
let m be Element of NAT ; ::_thesis: ( n <= m implies ||.((Sq . m) - Sq0).|| < p )
assume  n <= m ; ::_thesis: ||.((Sq . m) - Sq0).|| < p
then  abs ((||.(Sq - Sq0).|| . m) - 0) < p by A3;
then  abs ||.((Sq - Sq0) . m).|| < p by NORMSP_0:def_4;
then  abs ||.((Sq . m) - Sq0).|| < p by NORMSP_1:def_4;
hence  ||.((Sq . m) - Sq0).|| < p by ABSVALUE:def_1; ::_thesis: verum
end;
end;
hence  Sq is convergent by NORMSP_1:def_6; ::_thesis: lim Sq = Sq0
hence  lim Sq = Sq0 by A2, NORMSP_1:def_7; ::_thesis: verum
end;

theorem Th65: :: LPSPACE2:65
for X being RealNormSpace
for Sq being sequence of X st Sq is Cauchy_sequence_by_Norm holds
ex N being V167() sequence of NAT st
for i, j being Element of NAT st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i)
proof
let X be RealNormSpace; ::_thesis: for Sq being sequence of X st Sq is Cauchy_sequence_by_Norm holds
ex N being V167() sequence of NAT st
for i, j being Element of NAT st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i)
let Sq be sequence of X; ::_thesis: ( Sq is Cauchy_sequence_by_Norm implies ex N being V167() sequence of NAT st
for i, j being Element of NAT st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i) )
assume A1: Sq is Cauchy_sequence_by_Norm ; ::_thesis: ex N being V167() sequence of NAT st
for i, j being Element of NAT st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i)
 1 = 2 to_power (- 0) by POWER:24;
then consider N0 being Element of NAT  such that
A2: for j, i being Element of NAT st j >= N0 & i >= N0 holds
||.((Sq . j) - (Sq . i)).|| < 2 to_power (- 0) by A1, RSSPACE3:8;
defpred S1[ set , set , set ] means  ex n, x, y being Element of NAT st
( n = $1 & x = $2 & y = $3 & ( ( for j being Element of NAT st j >= x holds
||.((Sq . j) - (Sq . x)).|| < 2 to_power (- n) ) implies ( x < y & ( for j being Element of NAT st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) ) ) );
A3: for n, x being Element of NAT ex y being Element of NAT st S1[n,x,y]
proof
let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S1[n,x,y]
now__::_thesis:_(_(_for_j_being_Element_of_NAT_st_j_>=_x_holds_
||.((Sq_._j)_-_(Sq_._x)).||_<_2_to_power_(-_n)_)_implies_ex_y_being_Element_of_NAT_st_
(_x_<_y_&_(_for_j_being_Element_of_NAT_st_j_>=_y_holds_
||.((Sq_._j)_-_(Sq_._y)).||_<_2_to_power_(-_(n_+_1))_)_)_)
assume  for j being Element of NAT st j >= x holds
||.((Sq . j) - (Sq . x)).|| < 2 to_power (- n) ; ::_thesis: ex y being Element of NAT st
( x < y & ( for j being Element of NAT st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) )
 0 < 2 to_power (- (n + 1)) by POWER:34;
then consider N2 being Element of NAT  such that
A4: for j, i being Element of NAT st j >= N2 & i >= N2 holds
||.((Sq . j) - (Sq . i)).|| < 2 to_power (- (n + 1)) by A1, RSSPACE3:8;
set y = (max (x,N2)) + 1;
take y = (max (x,N2)) + 1; ::_thesis: ( x < y & ( for j being Element of NAT st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) )
 x <= max (x,N2) by XXREAL_0:25;
hence  x < y by NAT_1:13; ::_thesis: for j being Element of NAT st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1))
 N2 <= max (x,N2) by XXREAL_0:25;
then A5: N2 < y by NAT_1:13;
thus  for j being Element of NAT st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ::_thesis: verum
proof
let j be Element of NAT ; ::_thesis: ( j >= y implies ||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) )
assume  j >= y ; ::_thesis: ||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1))
then  ( j >= N2 & y >= N2 ) by A5, XXREAL_0:2;
hence  ||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) by A4; ::_thesis: verum
end;
end;
hence  ex y being Element of NAT st S1[n,x,y] ; ::_thesis: verum
end;
consider f being Function of NAT,NAT such that
A6: ( f . 0 = N0 & ( for n being Element of NAT holds S1[n,f . n,f . (n + 1)] ) ) from RECDEF_1:sch_2(A3);
defpred S2[ Element of NAT ] means  for j being Element of NAT st j >= f . $1 holds
||.((Sq . j) - (Sq . (f . $1))).|| < 2 to_power (- $1);
A7: S2[ 0 ] by A2, A6;
A8: now__::_thesis:_for_i_being_Element_of_NAT_st_S2[i]_holds_
S2[i_+_1]
let i be Element of NAT ; ::_thesis: ( S2[i] implies S2[i + 1] )
assume A9: S2[i] ; ::_thesis: S2[i + 1]
 ex n, x, y being Element of NAT st
( n = i & x = f . i & y = f . (i + 1) & ( ( for j being Element of NAT st j >= x holds
||.((Sq . j) - (Sq . x)).|| < 2 to_power (- n) ) implies ( x < y & ( for j being Element of NAT st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) ) ) ) by A6;
hence  S2[i + 1] by A9; ::_thesis: verum
end;
A10: for i being Element of NAT holds S2[i] from NAT_1:sch_1(A7, A8);
A11: f is Real_Sequence by FUNCT_2:7;
now__::_thesis:_for_i_being_Element_of_NAT_holds_f_._i_<_f_._(i_+_1)
let i be Element of NAT ; ::_thesis: f . i < f . (i + 1)
 ex n, x, y being Element of NAT st
( n = i & x = f . i & y = f . (i + 1) & ( ( for j being Element of NAT st j >= x holds
||.((Sq . j) - (Sq . x)).|| < 2 to_power (- n) ) implies ( x < y & ( for j being Element of NAT st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) ) ) ) by A6;
hence  f . i < f . (i + 1) by A10; ::_thesis: verum
end;
then  f is V167() by A11, SEQM_3:def_6;
hence  ex N being V167() sequence of NAT st
for i, j being Element of NAT st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i) by A10; ::_thesis: verum
end;

theorem Th66: :: LPSPACE2:66
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds
for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds
for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being positive Real
for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds
for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)
let M be sigma_Measure of S; ::_thesis: for k being positive Real
for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds
for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)
let k be positive Real; ::_thesis: for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds
for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)
let F be Functional_Sequence of X,REAL; ::_thesis: ( ( for m being Nat holds F . m in Lp_Functions (M,k) ) implies for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k) )
assume A1: for m being Nat holds F . m in Lp_Functions (M,k) ; ::_thesis: for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)
defpred S1[ Nat] means (Partial_Sums F) . $1 in Lp_Functions (M,k);
 (Partial_Sums F) . 0 = F . 0 by MESFUN9C:def_2;
then A2: S1[ 0 ] by A1;
A3: now__::_thesis:_for_j_being_Nat_st_S1[j]_holds_
S1[j_+_1]
let j be Nat; ::_thesis: ( S1[j] implies S1[j + 1] )
assume  S1[j] ; ::_thesis: S1[j + 1]
then A4: ( (Partial_Sums F) . j in Lp_Functions (M,k) & F . (j + 1) in Lp_Functions (M,k) ) by A1;
 (Partial_Sums F) . (j + 1) = ((Partial_Sums F) . j) + (F . (j + 1)) by MESFUN9C:def_2;
hence  S1[j + 1] by A4, Th25; ::_thesis: verum
end;
 for j being Nat holds S1[j] from NAT_1:sch_2(A2, A3);
hence  for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k) ; ::_thesis: verum
end;

theorem Th67: :: LPSPACE2:67
for X being non empty set
for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m is nonnegative ) holds
for m being Nat holds (Partial_Sums F) . m is nonnegative
proof
let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m is nonnegative ) holds
for m being Nat holds (Partial_Sums F) . m is nonnegative
let F be Functional_Sequence of X,REAL; ::_thesis: ( ( for m being Nat holds F . m is nonnegative ) implies for m being Nat holds (Partial_Sums F) . m is nonnegative )
assume A1: for m being Nat holds F . m is nonnegative ; ::_thesis: for m being Nat holds (Partial_Sums F) . m is nonnegative
defpred S1[ Nat] means (Partial_Sums F) . $1 is nonnegative ;
 (Partial_Sums F) . 0 = F . 0 by MESFUN9C:def_2;
then A2: S1[ 0 ] by A1;
A3: now__::_thesis:_for_k_being_Nat_st_S1[k]_holds_
S1[k_+_1]
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume  S1[k] ; ::_thesis: S1[k + 1]
then A4: ( (Partial_Sums F) . k is nonnegative & F . (k + 1) is nonnegative ) by A1;
 (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by MESFUN9C:def_2;
hence  S1[k + 1] by A4, MESFUNC6:56; ::_thesis: verum
end;
 for k being Nat holds S1[k] from NAT_1:sch_2(A2, A3);
hence  for m being Nat holds (Partial_Sums F) . m is nonnegative ; ::_thesis: verum
end;

theorem Th68: :: LPSPACE2:68
for X being non empty set
for F being Functional_Sequence of X,REAL
for x being Element of X
for n, m being Nat st F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m holds
((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x
proof
let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,REAL
for x being Element of X
for n, m being Nat st F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m holds
((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x
let F be Functional_Sequence of X,REAL; ::_thesis: for x being Element of X
for n, m being Nat st F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m holds
((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x
let x be Element of X; ::_thesis: for n, m being Nat st F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m holds
((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x
let n, m be Nat; ::_thesis: ( F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m implies ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x )
assume A1: F is with_the_same_dom ; ::_thesis: ( not x in dom (F . 0) or ex k being Nat st not F . k is nonnegative or not n <= m or ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x )
assume A2: x in dom (F . 0) ; ::_thesis: ( ex k being Nat st not F . k is nonnegative or not n <= m or ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x )
assume A3: for m being Nat holds F . m is nonnegative ; ::_thesis: ( not n <= m or ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x )
assume A4: n <= m ; ::_thesis: ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x
set PF = Partial_Sums F;
defpred S1[ Nat] means ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . $1) . x;
A5: for k being Nat holds ((Partial_Sums F) . k) . x <= ((Partial_Sums F) . (k + 1)) . x
proof
let k be Nat; ::_thesis: ((Partial_Sums F) . k) . x <= ((Partial_Sums F) . (k + 1)) . x
A6: (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by MESFUN9C:def_2;
A7: dom ((Partial_Sums F) . (k + 1)) = dom (F . 0) by A1, MESFUN9C:11;
 ( F . (k + 1) is nonnegative & (Partial_Sums F) . k is nonnegative ) by A3, Th67;
then  ( 0 <= (F . (k + 1)) . x & 0 <= ((Partial_Sums F) . k) . x ) by MESFUNC6:51;
then  (((Partial_Sums F) . k) . x) + 0 <= (((Partial_Sums F) . k) . x) + ((F . (k + 1)) . x) by XREAL_1:7;
hence  ((Partial_Sums F) . k) . x <= ((Partial_Sums F) . (k + 1)) . x by A7, A2, A6, VALUED_1:def_1; ::_thesis: verum
end;
A8: for k being Nat st k >= n & ( for l being Nat st l >= n & l < k holds
S1[l] ) holds
S1[k]
proof
let k be Nat; ::_thesis: ( k >= n & ( for l being Nat st l >= n & l < k holds
S1[l] ) implies S1[k] )
assume A9: ( k >= n & ( for l being Nat st l >= n & l < k holds
S1[l] ) ) ; ::_thesis: S1[k]
now__::_thesis:_(_k_>_n_implies_S1[k]_)
assume  k > n ; ::_thesis: S1[k]
then  k >= n + 1 by NAT_1:13;
then A10: ( k = n + 1 or k > n + 1 ) by XXREAL_0:1;
now__::_thesis:_(_k_>_n_+_1_implies_S1[k]_)
assume A11: k > n + 1 ; ::_thesis: S1[k]
then reconsider l = k - 1 as Element of NAT by NAT_1:20;
 k < k + 1 by NAT_1:13;
then  ( k > l & l >= n ) by A11, XREAL_1:19;
then A12: ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . l) . x by A9;
 k = l + 1 ;
then  ((Partial_Sums F) . l) . x <= ((Partial_Sums F) . k) . x by A5;
hence  S1[k] by A12, XXREAL_0:2; ::_thesis: verum
end;
hence  S1[k] by A10, A5; ::_thesis: verum
end;
hence  S1[k] by A9, XXREAL_0:1; ::_thesis: verum
end;
 for k being Nat st k >= n holds
S1[k] from NAT_1:sch_9(A8);
hence  ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x by A4; ::_thesis: verum
end;

theorem Th69: :: LPSPACE2:69
for X being non empty set
for F being Functional_Sequence of X,REAL st F is with_the_same_dom holds
abs F is with_the_same_dom
proof
let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,REAL st F is with_the_same_dom holds
abs F is with_the_same_dom
let F be Functional_Sequence of X,REAL; ::_thesis: ( F is with_the_same_dom implies abs F is with_the_same_dom )
assume A1: F is with_the_same_dom ; ::_thesis: abs F is with_the_same_dom
 for n, m being Nat holds dom ((abs F) . n) = dom ((abs F) . m)
proof
let n, m be Nat; ::_thesis: dom ((abs F) . n) = dom ((abs F) . m)
 ( (abs F) . n = abs (F . n) & (abs F) . m = abs (F . m) ) by SEQFUNC:def_4;
then  ( dom ((abs F) . n) = dom (F . n) & dom ((abs F) . m) = dom (F . m) ) by VALUED_1:def_11;
hence  dom ((abs F) . n) = dom ((abs F) . m) by A1, MESFUNC8:def_2; ::_thesis: verum
end;
hence  abs F is with_the_same_dom by MESFUNC8:def_2; ::_thesis: verum
end;

theorem Th70: :: LPSPACE2:70
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for k being geq_than_1 Real
for Sq being sequence of (Lp-Space (M,k)) st Sq is Cauchy_sequence_by_Norm holds
Sq is convergent
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being geq_than_1 Real
for Sq being sequence of (Lp-Space (M,k)) st Sq is Cauchy_sequence_by_Norm holds
Sq is convergent
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for k being geq_than_1 Real
for Sq being sequence of (Lp-Space (M,k)) st Sq is Cauchy_sequence_by_Norm holds
Sq is convergent
let M be sigma_Measure of S; ::_thesis: for k being geq_than_1 Real
for Sq being sequence of (Lp-Space (M,k)) st Sq is Cauchy_sequence_by_Norm holds
Sq is convergent
let k be geq_than_1 Real; ::_thesis: for Sq being sequence of (Lp-Space (M,k)) st Sq is Cauchy_sequence_by_Norm holds
Sq is convergent
let Sq be sequence of (Lp-Space (M,k)); ::_thesis: ( Sq is Cauchy_sequence_by_Norm implies Sq is convergent )
A1: 1 <= k by Def1;
assume A2: Sq is Cauchy_sequence_by_Norm ; ::_thesis: Sq is convergent
consider Fsq being with_the_same_dom Functional_Sequence of X,REAL such that
A3: for n being Element of NAT holds
( Fsq . n in Lp_Functions (M,k) & Fsq . n in Sq . n & Sq . n = a.e-eq-class_Lp ((Fsq . n),M,k) & ex r being Real st
( 0 <= r & r = Integral (M,((abs (Fsq . n)) to_power k)) & ||.(Sq . n).|| = r to_power (1 / k) ) ) by Th63;
 Fsq . 0 in Lp_Functions (M,k) by A3;
then A4: ex D being Element of S st
( M . (D `) = 0 & dom (Fsq . 0) = D & Fsq . 0 is_measurable_on D ) by Th35;
then reconsider E = dom (Fsq . 0) as Element of S ;
consider N being V167() sequence of NAT such that
A5: for i, j being Element of NAT st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i) by Th65, A2;
deffunc H1( Nat) ->  Element of bool [:X,REAL:] = Fsq . (N . $1);
consider F1 being Functional_Sequence of X,REAL such that
A6: for n being Nat holds F1 . n = H1(n) from SEQFUNC:sch_1();
A7: for n being Nat holds
( dom (F1 . n) = E & F1 . n in Lp_Functions (M,k) & F1 . n is_measurable_on E & abs (F1 . n) in Lp_Functions (M,k) )
proof
let n be Nat; ::_thesis: ( dom (F1 . n) = E & F1 . n in Lp_Functions (M,k) & F1 . n is_measurable_on E & abs (F1 . n) in Lp_Functions (M,k) )
A8: F1 . n = Fsq . (N . n) by A6;
hence A9: ( dom (F1 . n) = E & F1 . n in Lp_Functions (M,k) ) by A3, MESFUNC8:def_2; ::_thesis: ( F1 . n is_measurable_on E & abs (F1 . n) in Lp_Functions (M,k) )
then  ex F being PartFunc of X,REAL st
( F1 . n = F & ex ND being Element of S st
( M . (ND `) = 0 & dom F = ND & F is_measurable_on ND & (abs F) to_power k is_integrable_on M ) ) ;
hence  F1 . n is_measurable_on E by A8, MESFUNC8:def_2; ::_thesis: abs (F1 . n) in Lp_Functions (M,k)
thus  abs (F1 . n) in Lp_Functions (M,k) by A9, Th28; ::_thesis: verum
end;
 for n, m being Nat holds dom (F1 . n) = dom (F1 . m)
proof
let n, m be Nat; ::_thesis: dom (F1 . n) = dom (F1 . m)
 ( dom (F1 . n) = E & dom (F1 . m) = E ) by A7;
hence  dom (F1 . n) = dom (F1 . m) ; ::_thesis: verum
end;
then reconsider F1 = F1 as with_the_same_dom Functional_Sequence of X,REAL by MESFUNC8:def_2;
deffunc H2( Nat) ->  Element of bool [:X,REAL:] = (F1 . ($1 + 1)) - (F1 . $1);
consider FMF being Functional_Sequence of X,REAL such that
A10: for n being Nat holds FMF . n = H2(n) from SEQFUNC:sch_1();
A11: for n being Nat holds
( dom (FMF . n) = E & FMF . n in Lp_Functions (M,k) )
proof
let n be Nat; ::_thesis: ( dom (FMF . n) = E & FMF . n in Lp_Functions (M,k) )
A12: ( dom (F1 . n) = E & dom (F1 . (n + 1)) = E ) by A7;
 FMF . n = (F1 . (n + 1)) - (F1 . n) by A10;
then  dom (FMF . n) = (dom (F1 . (n + 1))) /\ (dom (F1 . n)) by VALUED_1:12;
hence  dom (FMF . n) = E by A12; ::_thesis: FMF . n in Lp_Functions (M,k)
 ( Fsq . (N . (n + 1)) in Lp_Functions (M,k) & Fsq . (N . n) in Lp_Functions (M,k) ) by A3;
then  ( F1 . (n + 1) in Lp_Functions (M,k) & F1 . n in Lp_Functions (M,k) ) by A6;
then  (F1 . (n + 1)) - (F1 . n) in Lp_Functions (M,k) by Th27;
hence  FMF . n in Lp_Functions (M,k) by A10; ::_thesis: verum
end;
 for n, m being Nat holds dom (FMF . n) = dom (FMF . m)
proof
let n, m be Nat; ::_thesis: dom (FMF . n) = dom (FMF . m)
 ( dom (FMF . n) = E & dom (FMF . m) = E ) by A11;
hence  dom (FMF . n) = dom (FMF . m) ; ::_thesis: verum
end;
then reconsider FMF = FMF as with_the_same_dom Functional_Sequence of X,REAL by MESFUNC8:def_2;
set AbsFMF = abs FMF;
A13: for n being Nat holds
( (abs FMF) . n is nonnegative & dom ((abs FMF) . n) = E & abs ((abs FMF) . n) = (abs FMF) . n & (abs FMF) . n in Lp_Functions (M,k) & (abs FMF) . n is_measurable_on E )
proof
let n be Nat; ::_thesis: ( (abs FMF) . n is nonnegative & dom ((abs FMF) . n) = E & abs ((abs FMF) . n) = (abs FMF) . n & (abs FMF) . n in Lp_Functions (M,k) & (abs FMF) . n is_measurable_on E )
A14: (abs FMF) . n = abs (FMF . n) by SEQFUNC:def_4;
hence  (abs FMF) . n is nonnegative ; ::_thesis: ( dom ((abs FMF) . n) = E & abs ((abs FMF) . n) = (abs FMF) . n & (abs FMF) . n in Lp_Functions (M,k) & (abs FMF) . n is_measurable_on E )
A15: ( dom (FMF . n) = E & FMF . n in Lp_Functions (M,k) ) by A11;
hence  ( dom ((abs FMF) . n) = E & abs ((abs FMF) . n) = (abs FMF) . n ) by A14, VALUED_1:def_11; ::_thesis: ( (abs FMF) . n in Lp_Functions (M,k) & (abs FMF) . n is_measurable_on E )
thus  (abs FMF) . n in Lp_Functions (M,k) by A11, A14, Th28; ::_thesis: (abs FMF) . n is_measurable_on E
then  ex D being Element of S st
( M . (D `) = 0 & dom ((abs FMF) . n) = D & (abs FMF) . n is_measurable_on D ) by Th35;
hence  (abs FMF) . n is_measurable_on E by A15, A14, VALUED_1:def_11; ::_thesis: verum
end;
reconsider AbsFMF = abs FMF as with_the_same_dom Functional_Sequence of X,REAL by Th69;
deffunc H3( Nat) ->  Element of bool [:X,REAL:] = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . $1);
consider G being Functional_Sequence of X,REAL such that
A16: for n being Nat holds G . n = H3(n) from SEQFUNC:sch_1();
A17: for n being Nat holds
( dom (G . n) = E & G . n in Lp_Functions (M,k) & G . n is nonnegative & G . n is_measurable_on E & abs (G . n) = G . n )
proof
let n be Nat; ::_thesis: ( dom (G . n) = E & G . n in Lp_Functions (M,k) & G . n is nonnegative & G . n is_measurable_on E & abs (G . n) = G . n )
A18: G . n = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . n) by A16;
then A19: dom (G . n) = (dom (abs (F1 . 0))) /\ (dom ((Partial_Sums AbsFMF) . n)) by VALUED_1:def_1
.= (dom (F1 . 0)) /\ (dom ((Partial_Sums AbsFMF) . n)) by VALUED_1:def_11
.= (dom (F1 . 0)) /\ (dom (AbsFMF . 0)) by MESFUN9C:11 ;
A20: ( (Partial_Sums AbsFMF) . n in Lp_Functions (M,k) & (Partial_Sums AbsFMF) . n is nonnegative & (Partial_Sums AbsFMF) . n is_measurable_on E ) by A13, Th66, Th67, MESFUN9C:16;
A21: dom (AbsFMF . 0) = E by A13;
A22: ( F1 . 0 in Lp_Functions (M,k) & dom (F1 . 0) = E & F1 . 0 is_measurable_on E ) by A7;
then  ( abs (F1 . 0) in Lp_Functions (M,k) & abs (F1 . 0) is nonnegative & abs (F1 . 0) is_measurable_on E ) by Th28, MESFUNC6:48;
hence  ( dom (G . n) = E & G . n in Lp_Functions (M,k) & G . n is nonnegative & G . n is_measurable_on E & abs (G . n) = G . n ) by A19, A22, A21, A18, A20, Th14, Th25, MESFUNC6:26, MESFUNC6:56; ::_thesis: verum
end;
deffunc H4( Nat) ->  Element of bool [:X,REAL:] = (G . $1) to_power k;
consider Gp being Functional_Sequence of X,REAL such that
A23: for n being Nat holds Gp . n = H4(n) from SEQFUNC:sch_1();
A24: for n being Nat holds
( (G . n) to_power k is nonnegative & (G . n) to_power k is_measurable_on E )
proof
let n be Nat; ::_thesis: ( (G . n) to_power k is nonnegative & (G . n) to_power k is_measurable_on E )
A25: G . n is nonnegative by A17;
hence  (G . n) to_power k is nonnegative ; ::_thesis: (G . n) to_power k is_measurable_on E
 ( G . n is_measurable_on E & dom (G . n) = E ) by A17;
hence  (G . n) to_power k is_measurable_on E by A25, MESFUN6C:29; ::_thesis: verum
end;
reconsider ExtGp = R_EAL Gp as Functional_Sequence of X,ExtREAL ;
A26: for n being Nat holds
( dom (ExtGp . n) = E & ExtGp . n is_measurable_on E & ExtGp . n is nonnegative )
proof
let n be Nat; ::_thesis: ( dom (ExtGp . n) = E & ExtGp . n is_measurable_on E & ExtGp . n is nonnegative )
 ExtGp . n = R_EAL ((G . n) to_power k) by A23;
then  dom (ExtGp . n) = dom (G . n) by MESFUN6C:def_4;
hence  dom (ExtGp . n) = E by A17; ::_thesis: ( ExtGp . n is_measurable_on E & ExtGp . n is nonnegative )
 (G . n) to_power k is_measurable_on E by A24;
then  R_EAL ((G . n) to_power k) is_measurable_on E by MESFUNC6:def_1;
hence  ExtGp . n is_measurable_on E by A23; ::_thesis: ExtGp . n is nonnegative
 (G . n) to_power k is nonnegative by A24;
hence  ExtGp . n is nonnegative by A23; ::_thesis: verum
end;
then A27: ( dom (ExtGp . 0) = E & ExtGp . 0 is nonnegative ) ;
 for n, m being Nat holds dom (ExtGp . n) = dom (ExtGp . m)
proof
let n, m be Nat; ::_thesis: dom (ExtGp . n) = dom (ExtGp . m)
 ( dom (ExtGp . n) = E & dom (ExtGp . m) = E ) by A26;
hence  dom (ExtGp . n) = dom (ExtGp . m) ; ::_thesis: verum
end;
then reconsider ExtGp = ExtGp as with_the_same_dom Functional_Sequence of X,ExtREAL by MESFUNC8:def_2;
A28: for n, m being Nat st n <= m holds
for x being Element of X st x in E holds
(ExtGp . n) . x <= (ExtGp . m) . x
proof
let n, m be Nat; ::_thesis: ( n <= m implies for x being Element of X st x in E holds
(ExtGp . n) . x <= (ExtGp . m) . x )
assume A29: n <= m ; ::_thesis: for x being Element of X st x in E holds
(ExtGp . n) . x <= (ExtGp . m) . x
let x be Element of X; ::_thesis: ( x in E implies (ExtGp . n) . x <= (ExtGp . m) . x )
assume A30: x in E ; ::_thesis: (ExtGp . n) . x <= (ExtGp . m) . x
then A31: ( x in dom (G . n) & x in dom (G . m) ) by A17;
then  ( x in dom ((G . n) to_power k) & x in dom ((G . m) to_power k) ) by MESFUN6C:def_4;
then  ( ((G . n) . x) to_power k = ((G . n) to_power k) . x & ((G . m) . x) to_power k = ((G . m) to_power k) . x ) by MESFUN6C:def_4;
then A32: ( ((G . n) . x) to_power k = (ExtGp . n) . x & ((G . m) . x) to_power k = (ExtGp . m) . x ) by A23;
 dom (AbsFMF . 0) = E by A13;
then  ((Partial_Sums AbsFMF) . n) . x <= ((Partial_Sums AbsFMF) . m) . x by Th68, A29, A30, A13;
then A33: ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . n) . x) <= ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . m) . x) by XREAL_1:6;
 ( G . m = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . m) & G . n = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . n) ) by A16;
then A34: ( (G . m) . x = ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . m) . x) & (G . n) . x = ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . n) . x) ) by A31, VALUED_1:def_1;
 G . n is nonnegative by A17;
then  0 <= (G . n) . x by MESFUNC6:51;
hence  (ExtGp . n) . x <= (ExtGp . m) . x by A32, A33, A34, HOLDER_1:3; ::_thesis: verum
end;
A35: for x being Element of X st x in E holds
ExtGp # x is V169()
proof
let x be Element of X; ::_thesis: ( x in E implies ExtGp # x is V169() )
assume A36: x in E ; ::_thesis: ExtGp # x is V169()
 for n, m being Element of NAT st m <= n holds
(ExtGp # x) . m <= (ExtGp # x) . n
proof
let n, m be Element of NAT ; ::_thesis: ( m <= n implies (ExtGp # x) . m <= (ExtGp # x) . n )
assume  m <= n ; ::_thesis: (ExtGp # x) . m <= (ExtGp # x) . n
then  (ExtGp . m) . x <= (ExtGp . n) . x by A28, A36;
then  (ExtGp # x) . m <= (ExtGp . n) . x by MESFUNC5:def_13;
hence  (ExtGp # x) . m <= (ExtGp # x) . n by MESFUNC5:def_13; ::_thesis: verum
end;
hence  ExtGp # x is V169() by RINFSUP2:7; ::_thesis: verum
end;
A37: for x being Element of X st x in E holds
ExtGp # x is convergent
proof
let x be Element of X; ::_thesis: ( x in E implies ExtGp # x is convergent )
assume  x in E ; ::_thesis: ExtGp # x is convergent
then  ExtGp # x is V169() by A35;
hence  ExtGp # x is convergent by RINFSUP2:37; ::_thesis: verum
end;
then consider I being ExtREAL_sequence such that
A38: ( ( for n being Nat holds I . n = Integral (M,(ExtGp . n)) ) & I is convergent & Integral (M,(lim ExtGp)) = lim I ) by A27, A26, A28, MESFUNC9:52;
now__::_thesis:_for_y_being_set_st_y_in_rng_I_holds_
y_in_REAL
let y be set ; ::_thesis: ( y in rng I implies y in REAL )
assume  y in rng I ; ::_thesis: y in REAL
then consider x being Element of NAT  such that
A39: y = I . x by FUNCT_2:113;
A40: y = Integral (M,(Gp . x)) by A39, A38;
 G . x = abs (G . x) by A17;
then A41: Gp . x = (abs (G . x)) to_power k by A23;
 G . x in Lp_Functions (M,k) by A17;
hence  y in REAL by A40, A41, Th49; ::_thesis: verum
end;
then  rng I c= REAL by TARSKI:def_3;
then reconsider Ir = I as Function of NAT,REAL by FUNCT_2:6;
deffunc H5( Nat) ->  Element of ExtREAL = Integral (M,((AbsFMF . $1) to_power k));
A42: for x being Element of NAT holds H5(x) is Element of REAL
proof
let x be Element of NAT ; ::_thesis: H5(x) is Element of REAL
 AbsFMF . x in Lp_Functions (M,k) by A13;
then  Integral (M,((abs (AbsFMF . x)) to_power k)) in REAL by Th49;
hence  H5(x) is Element of REAL by A13; ::_thesis: verum
end;
consider KPAbsFMF being Function of NAT,REAL such that
A43: for x being Element of NAT holds KPAbsFMF . x = H5(x) from FUNCT_2:sch_9(A42);
deffunc H6( Nat) ->  Element of REAL = (KPAbsFMF . $1) to_power (1 / k);
A44: for x being Element of NAT holds H6(x) is Element of REAL ;
consider PAbsFMF being Function of NAT,REAL such that
A45: for x being Element of NAT holds PAbsFMF . x = H6(x) from FUNCT_2:sch_9(A44);
 F1 . 0 in Lp_Functions (M,k) by A7;
then reconsider RF0 = Integral (M,((abs (F1 . 0)) to_power k)) as Element of REAL by Th49;
deffunc H7( Element of NAT ) ->  Element of REAL = (RF0 to_power (1 / k)) + ((Partial_Sums PAbsFMF) . $1);
A46: for x being Element of NAT holds H7(x) is Element of REAL ;
consider QAbsFMF being Function of NAT,REAL such that
A47: for x being Element of NAT holds QAbsFMF . x = H7(x) from FUNCT_2:sch_9(A46);
A48: for n being Element of NAT holds (Ir . n) to_power (1 / k) <= QAbsFMF . n
proof
defpred S1[ Nat] means (Ir . $1) to_power (1 / k) <= QAbsFMF . $1;
A49: ( abs (F1 . 0) in Lp_Functions (M,k) & AbsFMF . 0 in Lp_Functions (M,k) ) by A13, A7;
 G . 0 = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . 0) by A16;
then A50: G . 0 = (abs (F1 . 0)) + (AbsFMF . 0) by MESFUN9C:def_2;
 Ir . 0 = Integral (M,(Gp . 0)) by A38;
then  Ir . 0 = Integral (M,((G . 0) to_power k)) by A23;
then A51: Ir . 0 = Integral (M,((abs ((abs (F1 . 0)) + (AbsFMF . 0))) to_power k)) by A17, A50;
 KPAbsFMF . 0 = Integral (M,((AbsFMF . 0) to_power k)) by A43;
then A52: KPAbsFMF . 0 = Integral (M,((abs (AbsFMF . 0)) to_power k)) by A13;
A53: RF0 = Integral (M,((abs (abs (F1 . 0))) to_power k)) ;
 QAbsFMF . 0 = (RF0 to_power (1 / k)) + ((Partial_Sums PAbsFMF) . 0) by A47;
then  QAbsFMF . 0 = (RF0 to_power (1 / k)) + (PAbsFMF . 0) by SERIES_1:def_1;
then  QAbsFMF . 0 = (RF0 to_power (1 / k)) + ((KPAbsFMF . 0) to_power (1 / k)) by A45;
then A54: S1[ 0 ] by A1, A49, A51, A52, A53, Th61;
A55: now__::_thesis:_for_n_being_Element_of_NAT_st_S1[n]_holds_
S1[n_+_1]
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume  S1[n] ; ::_thesis: S1[n + 1]
then A56: ((Ir . n) to_power (1 / k)) + (PAbsFMF . (n + 1)) <= (QAbsFMF . n) + (PAbsFMF . (n + 1)) by XREAL_1:6;
G . (n + 1) = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . (n + 1)) by A16
.= (abs (F1 . 0)) + (((Partial_Sums AbsFMF) . n) + (AbsFMF . (n + 1))) by MESFUN9C:def_2
.= ((abs (F1 . 0)) + ((Partial_Sums AbsFMF) . n)) + (AbsFMF . (n + 1)) by RFUNCT_1:8 ;
then A57: G . (n + 1) = (G . n) + (AbsFMF . (n + 1)) by A16;
A58: ( AbsFMF . (n + 1) in Lp_Functions (M,k) & G . n in Lp_Functions (M,k) ) by A13, A17;
 KPAbsFMF . (n + 1) = Integral (M,((AbsFMF . (n + 1)) to_power k)) by A43;
then A59: KPAbsFMF . (n + 1) = Integral (M,((abs (AbsFMF . (n + 1))) to_power k)) by A13;
A60: PAbsFMF . (n + 1) = (KPAbsFMF . (n + 1)) to_power (1 / k) by A45;
 ( Ir . n = Integral (M,(Gp . n)) & Ir . (n + 1) = Integral (M,(Gp . (n + 1))) ) by A38;
then  ( Ir . n = Integral (M,((G . n) to_power k)) & Ir . (n + 1) = Integral (M,((G . (n + 1)) to_power k)) ) by A23;
then  ( Ir . n = Integral (M,((abs (G . n)) to_power k)) & Ir . (n + 1) = Integral (M,((abs ((G . n) + (AbsFMF . (n + 1)))) to_power k)) ) by A57, A17;
then  (Ir . (n + 1)) to_power (1 / k) <= ((Ir . n) to_power (1 / k)) + (PAbsFMF . (n + 1)) by A1, A58, A59, A60, Th61;
then A61: (Ir . (n + 1)) to_power (1 / k) <= (QAbsFMF . n) + (PAbsFMF . (n + 1)) by A56, XXREAL_0:2;
(QAbsFMF . n) + (PAbsFMF . (n + 1)) = ((RF0 to_power (1 / k)) + ((Partial_Sums PAbsFMF) . n)) + (PAbsFMF . (n + 1)) by A47
.= (RF0 to_power (1 / k)) + (((Partial_Sums PAbsFMF) . n) + (PAbsFMF . (n + 1)))
.= (RF0 to_power (1 / k)) + ((Partial_Sums PAbsFMF) . (n + 1)) by SERIES_1:def_1 ;
hence  S1[n + 1] by A61, A47; ::_thesis: verum
end;
 for n being Element of NAT holds S1[n] from NAT_1:sch_1(A54, A55);
hence  for n being Element of NAT holds (Ir . n) to_power (1 / k) <= QAbsFMF . n ; ::_thesis: verum
end;
A62: for n being Element of NAT holds PAbsFMF . n = ||.((Sq . (N . (n + 1))) - (Sq . (N . n))).||
proof
let n be Element of NAT ; ::_thesis: PAbsFMF . n = ||.((Sq . (N . (n + 1))) - (Sq . (N . n))).||
set m = N . n;
set m1 = N . (n + 1);
A63: ( F1 . (n + 1) = Fsq . (N . (n + 1)) & F1 . n = Fsq . (N . n) ) by A6;
 AbsFMF . n = abs (FMF . n) by SEQFUNC:def_4;
then A64: AbsFMF . n = abs ((Fsq . (N . (n + 1))) - (Fsq . (N . n))) by A63, A10;
A65: ( Fsq . (N . (n + 1)) in Lp_Functions (M,k) & Fsq . (N . (n + 1)) in Sq . (N . (n + 1)) & Fsq . (N . n) in Lp_Functions (M,k) & Fsq . (N . n) in Sq . (N . n) ) by A3;
then  (- 1) (#) (Fsq . (N . n)) in (- 1) * (Sq . (N . n)) by Th54;
then  (Fsq . (N . (n + 1))) - (Fsq . (N . n)) in (Sq . (N . (n + 1))) + ((- 1) * (Sq . (N . n))) by Th54, A65;
then  (Fsq . (N . (n + 1))) - (Fsq . (N . n)) in (Sq . (N . (n + 1))) - (Sq . (N . n)) by RLVECT_1:16;
then A66: ex r being Real st
( 0 <= r & r = Integral (M,((abs ((Fsq . (N . (n + 1))) - (Fsq . (N . n)))) to_power k)) & ||.((Sq . (N . (n + 1))) - (Sq . (N . n))).|| = r to_power (1 / k) ) by Th53;
 PAbsFMF . n = (KPAbsFMF . n) to_power (1 / k) by A45;
hence  PAbsFMF . n = ||.((Sq . (N . (n + 1))) - (Sq . (N . n))).|| by A66, A64, A43; ::_thesis: verum
end;
 1 / 2 < 1 ;
then  abs (1 / 2) < 1 by ABSVALUE:def_1;
then A67: ( (1 / 2) GeoSeq is summable & Sum ((1 / 2) GeoSeq) = 1 / (1 - (1 / 2)) ) by SERIES_1:24;
 for n being Element of NAT holds
( 0 <= PAbsFMF . n & PAbsFMF . n <= ((1 / 2) GeoSeq) . n )
proof
let n be Element of NAT ; ::_thesis: ( 0 <= PAbsFMF . n & PAbsFMF . n <= ((1 / 2) GeoSeq) . n )
A68: PAbsFMF . n = ||.((Sq . (N . (n + 1))) - (Sq . (N . n))).|| by A62;
hence  0 <= PAbsFMF . n ; ::_thesis: PAbsFMF . n <= ((1 / 2) GeoSeq) . n
((1 / 2) GeoSeq) . n = (1 / 2) |^ n by PREPOWER:def_1
.= (1 / 2) to_power n by POWER:41 ;
then A69: ((1 / 2) GeoSeq) . n = 2 to_power (- n) by POWER:32;
 N is Real_Sequence by FUNCT_2:7;
then  N . n < N . (n + 1) by SEQM_3:def_6;
hence  PAbsFMF . n <= ((1 / 2) GeoSeq) . n by A5, A68, A69; ::_thesis: verum
end;
then  ( PAbsFMF is summable & Sum PAbsFMF <= Sum ((1 / 2) GeoSeq) ) by A67, SERIES_1:20;
then  Partial_Sums PAbsFMF is convergent by SERIES_1:def_2;
then  Partial_Sums PAbsFMF is bounded ;
then consider Br being real number  such that
A70: for n being Element of NAT holds (Partial_Sums PAbsFMF) . n < Br by SEQ_2:def_3;
 for n being Element of NAT holds Ir . n < ((RF0 to_power (1 / k)) + Br) to_power k
proof
let n be Element of NAT ; ::_thesis: Ir . n < ((RF0 to_power (1 / k)) + Br) to_power k
 (Ir . n) to_power (1 / k) <= QAbsFMF . n by A48;
then A71: (Ir . n) to_power (1 / k) <= (RF0 to_power (1 / k)) + ((Partial_Sums PAbsFMF) . n) by A47;
 (RF0 to_power (1 / k)) + ((Partial_Sums PAbsFMF) . n) < (RF0 to_power (1 / k)) + Br by A70, XREAL_1:8;
then A72: (Ir . n) to_power (1 / k) < (RF0 to_power (1 / k)) + Br by A71, XXREAL_0:2;
 Ir . n = Integral (M,(Gp . n)) by A38;
then  Ir . n = Integral (M,((G . n) to_power k)) by A23;
then A73: Ir . n = Integral (M,((abs (G . n)) to_power k)) by A17;
A74: G . n in Lp_Functions (M,k) by A17;
then  0 <= (Ir . n) to_power (1 / k) by Th49, A73, Th4;
then  ((Ir . n) to_power (1 / k)) to_power k < ((RF0 to_power (1 / k)) + Br) to_power k by A72, Th3;
then  (Ir . n) to_power ((1 / k) * k) < ((RF0 to_power (1 / k)) + Br) to_power k by A74, Th49, A73, HOLDER_1:2;
then  (Ir . n) to_power 1 < ((RF0 to_power (1 / k)) + Br) to_power k by XCMPLX_1:106;
hence  Ir . n < ((RF0 to_power (1 / k)) + Br) to_power k by POWER:25; ::_thesis: verum
end;
then A75: Ir is bounded_above by SEQ_2:def_3;
 for n, m being Element of NAT st n <= m holds
Ir . n <= Ir . m
proof
let n, m be Element of NAT ; ::_thesis: ( n <= m implies Ir . n <= Ir . m )
assume  n <= m ; ::_thesis: Ir . n <= Ir . m
then A76: for x being Element of X st x in E holds
(ExtGp . n) . x <= (ExtGp . m) . x by A28;
A77: ( ExtGp . n is_measurable_on E & ExtGp . m is_measurable_on E & ExtGp . n is nonnegative & ExtGp . m is nonnegative ) by A26;
A78: ( dom (ExtGp . n) = E & dom (ExtGp . m) = E ) by A26;
then A79: ( (ExtGp . n) | E = ExtGp . n & (ExtGp . m) | E = ExtGp . m ) by RELAT_1:68;
 ( I . n = Integral (M,(ExtGp . n)) & I . m = Integral (M,(ExtGp . m)) ) by A38;
hence  Ir . n <= Ir . m by A76, A78, A77, A79, MESFUNC9:15; ::_thesis: verum
end;
then  Ir is V169() by SEQM_3:6;
then A80: ( I is convergent_to_finite_number & lim I = lim Ir ) by A75, RINFSUP2:14;
reconsider LExtGp = lim ExtGp as PartFunc of X,ExtREAL ;
A81: ( E = dom LExtGp & LExtGp is_measurable_on E ) by A26, A27, A37, MESFUNC8:25, MESFUNC8:def_9;
A82: for x being set st x in dom LExtGp holds
0 <= LExtGp . x
proof
let x be set ; ::_thesis: ( x in dom LExtGp implies 0 <= LExtGp . x )
assume A83: x in dom LExtGp ; ::_thesis: 0 <= LExtGp . x
then reconsider x1 = x as Element of X ;
A84: x1 in E by A27, A83, MESFUNC8:def_9;
now__::_thesis:_for_k1_being_Nat_holds_0_<=_(ExtGp_#_x1)_._k1
let k1 be Nat; ::_thesis: 0 <= (ExtGp # x1) . k1
reconsider k = k1 as Element of NAT by ORDINAL1:def_12;
 ExtGp # x1 is V169() by A35, A84;
then A85: (ExtGp # x1) . 0 <= (ExtGp # x1) . k by RINFSUP2:7;
 0 <= (ExtGp . 0) . x1 by A27, SUPINF_2:39;
hence  0 <= (ExtGp # x1) . k1 by A85, MESFUNC5:def_13; ::_thesis: verum
end;
then  0 <= lim (ExtGp # x1) by A84, A37, MESFUNC9:10;
hence  0 <= LExtGp . x by A83, MESFUNC8:def_9; ::_thesis: verum
end;
A86: eq_dom (LExtGp,+infty) = E /\ (eq_dom (LExtGp,+infty)) by A81, RELAT_1:132, XBOOLE_1:28;
then reconsider EE = eq_dom (LExtGp,+infty) as Element of S by A81, MESFUNC1:33;
reconsider E0 = E \ EE as Element of S ;
 E0 ` = (X \ E) \/ (X /\ EE) by XBOOLE_1:52;
then A87: E0 ` = (E `) \/ EE by XBOOLE_1:28;
 M . EE = 0 by A38, A80, A81, A82, A86, MESFUNC9:13, SUPINF_2:52;
then A88: EE is measure_zero of M by MEASURE1:def_7;
 E ` is Element of S by MEASURE1:34;
then  E ` is measure_zero of M by A4, MEASURE1:def_7;
then  E0 ` is measure_zero of M by A87, A88, MEASURE1:37;
then A89: M . (E0 `) = 0 by MEASURE1:def_7;
A90: for x being Element of X st x in E0 holds
LExtGp . x in REAL
proof
let x be Element of X; ::_thesis: ( x in E0 implies LExtGp . x in REAL )
assume  x in E0 ; ::_thesis: LExtGp . x in REAL
then  ( x in E & not x in EE ) by XBOOLE_0:def_5;
then  ( LExtGp . x <> +infty & 0 <= LExtGp . x ) by A81, A82, MESFUNC1:def_15;
hence  LExtGp . x in REAL by XXREAL_0:14; ::_thesis: verum
end;
A91: for x being Element of X st x in E0 holds
( Gp # x is convergent & lim (Gp # x) = lim (ExtGp # x) )
proof
let x be Element of X; ::_thesis: ( x in E0 implies ( Gp # x is convergent & lim (Gp # x) = lim (ExtGp # x) ) )
assume A92: x in E0 ; ::_thesis: ( Gp # x is convergent & lim (Gp # x) = lim (ExtGp # x) )
then A93: x in E by XBOOLE_0:def_5;
then  LExtGp . x = lim (ExtGp # x) by A81, MESFUNC8:def_9;
then A94: lim (ExtGp # x) in REAL by A90, A92;
 ExtGp # x is convergent by A37, A93;
then A95: ex g being real number st
( lim (ExtGp # x) = g & ( for p being real number st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.(((ExtGp # x) . m) - (lim (ExtGp # x))).| < p ) & ExtGp # x is convergent_to_finite_number ) by A94, MESFUNC5:def_12;
 ExtGp # x = Gp # x by MESFUN7C:1;
hence  ( Gp # x is convergent & lim (Gp # x) = lim (ExtGp # x) ) by A95, RINFSUP2:15; ::_thesis: verum
end;
A96: for x being Element of X st x in E0 holds
for n being Element of NAT holds (Gp # x) . n = ((G # x) . n) to_power k
proof
let x be Element of X; ::_thesis: ( x in E0 implies for n being Element of NAT holds (Gp # x) . n = ((G # x) . n) to_power k )
assume A97: x in E0 ; ::_thesis: for n being Element of NAT holds (Gp # x) . n = ((G # x) . n) to_power k
hereby  ::_thesis: verum
let n be Element of NAT ; ::_thesis: (Gp # x) . n = ((G # x) . n) to_power k
 x in E by A97, XBOOLE_0:def_5;
then  x in dom (G . n) by A17;
then A98: x in dom ((G . n) to_power k) by MESFUN6C:def_4;
(Gp # x) . n = (Gp . n) . x by SEQFUNC:def_10
.= ((G . n) to_power k) . x by A23
.= ((G . n) . x) to_power k by A98, MESFUN6C:def_4 ;
hence  (Gp # x) . n = ((G # x) . n) to_power k by SEQFUNC:def_10; ::_thesis: verum
end;
end;
A99: for x being Element of X st x in E0 holds
(Partial_Sums AbsFMF) # x is convergent
proof
let x be Element of X; ::_thesis: ( x in E0 implies (Partial_Sums AbsFMF) # x is convergent )
assume A100: x in E0 ; ::_thesis: (Partial_Sums AbsFMF) # x is convergent
then A101: Gp # x is convergent by A91;
A102: now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<=_(G_#_x)_._n
let n be Element of NAT ; ::_thesis: 0 <= (G # x) . n
 G . n is nonnegative by A17;
then  0 <= (G . n) . x by MESFUNC6:51;
hence  0 <= (G # x) . n by SEQFUNC:def_10; ::_thesis: verum
end;
 for n being Element of NAT holds (Gp # x) . n = ((G # x) . n) to_power k by A100, A96;
then A103: G # x is convergent by A101, A102, Th9;
now__::_thesis:_for_s_being_real_number_st_0_<_s_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((((Partial_Sums_AbsFMF)_#_x)_._m)_-_(((Partial_Sums_AbsFMF)_#_x)_._n))_<_s
let s be real number ; ::_thesis: ( 0 < s implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((Partial_Sums AbsFMF) # x) . m) - (((Partial_Sums AbsFMF) # x) . n)) < s )
assume  0 < s ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((Partial_Sums AbsFMF) # x) . m) - (((Partial_Sums AbsFMF) # x) . n)) < s
then consider n being Element of NAT  such that
A104: for m being Element of NAT st n <= m holds
abs (((G # x) . m) - ((G # x) . n)) < s by A103, SEQ_4:41;
now__::_thesis:_for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((((Partial_Sums_AbsFMF)_#_x)_._m)_-_(((Partial_Sums_AbsFMF)_#_x)_._n))_<_s
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((((Partial_Sums AbsFMF) # x) . m) - (((Partial_Sums AbsFMF) # x) . n)) < s )
assume A105: n <= m ; ::_thesis: abs ((((Partial_Sums AbsFMF) # x) . m) - (((Partial_Sums AbsFMF) # x) . n)) < s
 x in E by A100, XBOOLE_0:def_5;
then A106: ( x in dom (G . n) & x in dom (G . m) ) by A17;
 ( G . m = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . m) & G . n = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . n) ) by A16;
then  ( (G . m) . x = ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . m) . x) & (G . n) . x = ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . n) . x) ) by A106, VALUED_1:def_1;
then  ( (G # x) . m = ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . m) . x) & (G # x) . n = ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . n) . x) ) by SEQFUNC:def_10;
then A107: ((G # x) . m) - ((G # x) . n) = (((Partial_Sums AbsFMF) . m) . x) - (((Partial_Sums AbsFMF) . n) . x) ;
 ( ((Partial_Sums AbsFMF) # x) . m = ((Partial_Sums AbsFMF) . m) . x & ((Partial_Sums AbsFMF) # x) . n = ((Partial_Sums AbsFMF) . n) . x ) by SEQFUNC:def_10;
hence  abs ((((Partial_Sums AbsFMF) # x) . m) - (((Partial_Sums AbsFMF) # x) . n)) < s by A104, A105, A107; ::_thesis: verum
end;
hence  ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((Partial_Sums AbsFMF) # x) . m) - (((Partial_Sums AbsFMF) # x) . n)) < s ; ::_thesis: verum
end;
hence  (Partial_Sums AbsFMF) # x is convergent by SEQ_4:41; ::_thesis: verum
end;
A108: for x being Element of X st x in E0 holds
Partial_Sums (abs (FMF # x)) = (Partial_Sums AbsFMF) # x
proof
let x be Element of X; ::_thesis: ( x in E0 implies Partial_Sums (abs (FMF # x)) = (Partial_Sums AbsFMF) # x )
assume  x in E0 ; ::_thesis: Partial_Sums (abs (FMF # x)) = (Partial_Sums AbsFMF) # x
then A109: x in E by XBOOLE_0:def_5;
defpred S1[ Nat] means (Partial_Sums (abs (FMF # x))) . $1 = ((Partial_Sums AbsFMF) # x) . $1;
(Partial_Sums (abs (FMF # x))) . 0 = (abs (FMF # x)) . 0 by SERIES_1:def_1
.= abs ((FMF # x) . 0) by VALUED_1:18
.= abs ((FMF . 0) . x) by SEQFUNC:def_10
.= (abs (FMF . 0)) . x by VALUED_1:18
.= (AbsFMF . 0) . x by SEQFUNC:def_4
.= ((Partial_Sums AbsFMF) . 0) . x by MESFUN9C:def_2
.= ((Partial_Sums AbsFMF) # x) . 0 by SEQFUNC:def_10 ;
then A110: S1[ 0 ] ;
A111: now__::_thesis:_for_n_being_Element_of_NAT_st_S1[n]_holds_
S1[n_+_1]
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A112: S1[n] ; ::_thesis: S1[n + 1]
A113: (Partial_Sums AbsFMF) . (n + 1) = ((Partial_Sums AbsFMF) . n) + (AbsFMF . (n + 1)) by MESFUN9C:def_2;
 dom (AbsFMF . 0) = E by A13;
then A114: x in dom ((Partial_Sums AbsFMF) . (n + 1)) by A109, MESFUN9C:11;
A115: (abs (FMF # x)) . (n + 1) = abs ((FMF # x) . (n + 1)) by VALUED_1:18
.= abs ((FMF . (n + 1)) . x) by SEQFUNC:def_10
.= (abs (FMF . (n + 1))) . x by VALUED_1:18
.= (AbsFMF . (n + 1)) . x by SEQFUNC:def_4 ;
(Partial_Sums (abs (FMF # x))) . (n + 1) = ((Partial_Sums (abs (FMF # x))) . n) + ((abs (FMF # x)) . (n + 1)) by SERIES_1:def_1
.= (((Partial_Sums AbsFMF) . n) . x) + ((AbsFMF . (n + 1)) . x) by A112, A115, SEQFUNC:def_10
.= ((Partial_Sums AbsFMF) . (n + 1)) . x by A113, A114, VALUED_1:def_1
.= ((Partial_Sums AbsFMF) # x) . (n + 1) by SEQFUNC:def_10 ;
hence  S1[n + 1] ; ::_thesis: verum
end;
 for n being Element of NAT holds S1[n] from NAT_1:sch_1(A110, A111);
hence  Partial_Sums (abs (FMF # x)) = (Partial_Sums AbsFMF) # x by FUNCT_2:63; ::_thesis: verum
end;
A116: for x being Element of X st x in E0 holds
for n being Element of NAT holds (F1 # x) . (n + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n)
proof
let x be Element of X; ::_thesis: ( x in E0 implies for n being Element of NAT holds (F1 # x) . (n + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n) )
assume  x in E0 ; ::_thesis: for n being Element of NAT holds (F1 # x) . (n + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n)
then A117: x in E by XBOOLE_0:def_5;
defpred S1[ Nat] means (F1 # x) . ($1 + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . $1);
 dom (FMF . 0) = E by A11;
then A118: x in dom ((F1 . (0 + 1)) - (F1 . 0)) by A10, A117;
(Partial_Sums (FMF # x)) . 0 = (FMF # x) . 0 by SERIES_1:def_1
.= (FMF . 0) . x by SEQFUNC:def_10
.= ((F1 . (0 + 1)) - (F1 . 0)) . x by A10 ;
then A119: (Partial_Sums (FMF # x)) . 0 = ((F1 . (0 + 1)) . x) - ((F1 . 0) . x) by A118, VALUED_1:13;
 ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . 0) = ((F1 . 0) . x) + ((Partial_Sums (FMF # x)) . 0) by SEQFUNC:def_10;
then A120: S1[ 0 ] by A119, SEQFUNC:def_10;
A121: now__::_thesis:_for_n_being_Element_of_NAT_st_S1[n]_holds_
S1[n_+_1]
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A122: S1[n] ; ::_thesis: S1[n + 1]
 dom (FMF . (n + 1)) = E by A11;
then A123: x in dom ((F1 . ((n + 1) + 1)) - (F1 . (n + 1))) by A10, A117;
(FMF # x) . (n + 1) = (FMF . (n + 1)) . x by SEQFUNC:def_10
.= ((F1 . ((n + 1) + 1)) - (F1 . (n + 1))) . x by A10 ;
then A124: (FMF # x) . (n + 1) = ((F1 . ((n + 1) + 1)) . x) - ((F1 . (n + 1)) . x) by A123, VALUED_1:13;
((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . (n + 1)) = ((F1 # x) . 0) + (((Partial_Sums (FMF # x)) . n) + ((FMF # x) . (n + 1))) by SERIES_1:def_1
.= (((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n)) + ((FMF # x) . (n + 1))
.= ((F1 . (n + 1)) . x) + ((FMF # x) . (n + 1)) by A122, SEQFUNC:def_10 ;
hence  S1[n + 1] by A124, SEQFUNC:def_10; ::_thesis: verum
end;
 for n being Element of NAT holds S1[n] from NAT_1:sch_1(A120, A121);
hence  for n being Element of NAT holds (F1 # x) . (n + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n) ; ::_thesis: verum
end;
A125: for x being Element of X st x in E0 holds
F1 # x is convergent
proof
let x be Element of X; ::_thesis: ( x in E0 implies F1 # x is convergent )
assume A126: x in E0 ; ::_thesis: F1 # x is convergent
then  Partial_Sums (abs (FMF # x)) = (Partial_Sums AbsFMF) # x by A108;
then  Partial_Sums (abs (FMF # x)) is convergent by A126, A99;
then  abs (FMF # x) is summable by SERIES_1:def_2;
then  FMF # x is absolutely_summable by SERIES_1:def_4;
then  FMF # x is summable ;
then A127: Partial_Sums (FMF # x) is convergent by SERIES_1:def_2;
now__::_thesis:_for_s_being_real_number_st_0_<_s_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_(((F1_#_x)_._m)_-_((F1_#_x)_._n))_<_s
let s be real number ; ::_thesis: ( 0 < s implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((F1 # x) . m) - ((F1 # x) . n)) < s )
assume  0 < s ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((F1 # x) . m) - ((F1 # x) . n)) < s
then consider n being Element of NAT  such that
A128: for m being Element of NAT st n <= m holds
abs (((Partial_Sums (FMF # x)) . m) - ((Partial_Sums (FMF # x)) . n)) < s by A127, SEQ_4:41;
set n1 = n + 1;
now__::_thesis:_for_m1_being_Element_of_NAT_st_n_+_1_<=_m1_holds_
abs_(((F1_#_x)_._m1)_-_((F1_#_x)_._(n_+_1)))_<_s
let m1 be Element of NAT ; ::_thesis: ( n + 1 <= m1 implies abs (((F1 # x) . m1) - ((F1 # x) . (n + 1))) < s )
assume A129: n + 1 <= m1 ; ::_thesis: abs (((F1 # x) . m1) - ((F1 # x) . (n + 1))) < s
 1 <= n + 1 by NAT_1:11;
then reconsider m = m1 - 1 as Element of NAT by A129, NAT_1:21, XXREAL_0:2;
 (n + 1) - 1 <= m1 - 1 by A129, XREAL_1:9;
then A130: abs (((Partial_Sums (FMF # x)) . m) - ((Partial_Sums (FMF # x)) . n)) < s by A128;
 m1 = m + 1 ;
then  ( (F1 # x) . (n + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n) & (F1 # x) . m1 = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . m) ) by A116, A126;
hence  abs (((F1 # x) . m1) - ((F1 # x) . (n + 1))) < s by A130; ::_thesis: verum
end;
hence  ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((F1 # x) . m) - ((F1 # x) . n)) < s ; ::_thesis: verum
end;
hence  F1 # x is convergent by SEQ_4:41; ::_thesis: verum
end;
set F2 = F1 || E0;
A131: for x being Element of X st x in E0 holds
(F1 || E0) # x is convergent
proof
let x be Element of X; ::_thesis: ( x in E0 implies (F1 || E0) # x is convergent )
assume A132: x in E0 ; ::_thesis: (F1 || E0) # x is convergent
then  F1 # x is convergent by A125;
hence  (F1 || E0) # x is convergent by A132, MESFUN9C:1; ::_thesis: verum
end;
A133: for x being Element of X st x in E0 holds
(F1 || E0) # x = F1 # x
proof
let x be Element of X; ::_thesis: ( x in E0 implies (F1 || E0) # x = F1 # x )
assume A134: x in E0 ; ::_thesis: (F1 || E0) # x = F1 # x
now__::_thesis:_for_n_being_Element_of_NAT_holds_((F1_||_E0)_#_x)_._n_=_(F1_#_x)_._n
let n be Element of NAT ; ::_thesis: ((F1 || E0) # x) . n = (F1 # x) . n
((F1 || E0) # x) . n = ((F1 || E0) . n) . x by SEQFUNC:def_10
.= ((F1 . n) | E0) . x by MESFUN9C:def_1
.= (F1 . n) . x by A134, FUNCT_1:49 ;
hence  ((F1 || E0) # x) . n = (F1 # x) . n by SEQFUNC:def_10; ::_thesis: verum
end;
hence  (F1 || E0) # x = F1 # x by FUNCT_2:63; ::_thesis: verum
end;
A135: for n being Nat holds
( dom ((F1 || E0) . n) = E0 & (F1 || E0) . n is_measurable_on E0 )
proof
let n be Nat; ::_thesis: ( dom ((F1 || E0) . n) = E0 & (F1 || E0) . n is_measurable_on E0 )
A136: dom (F1 . 0) = E by A7;
 dom ((F1 || E0) . n) = dom ((F1 . n) | E0) by MESFUN9C:def_1;
then  dom ((F1 || E0) . n) = (dom (F1 . n)) /\ E0 by RELAT_1:61;
then  dom ((F1 || E0) . n) = E /\ E0 by A7;
hence  dom ((F1 || E0) . n) = E0 by XBOOLE_1:28, XBOOLE_1:36; ::_thesis: (F1 || E0) . n is_measurable_on E0
 for m being Nat holds F1 . m is_measurable_on E0
proof
let m be Nat; ::_thesis: F1 . m is_measurable_on E0
 F1 . m is_measurable_on E by A7;
hence  F1 . m is_measurable_on E0 by MESFUNC6:16, XBOOLE_1:36; ::_thesis: verum
end;
hence  (F1 || E0) . n is_measurable_on E0 by A136, MESFUN9C:4, XBOOLE_1:36; ::_thesis: verum
end;
reconsider F2 = F1 || E0 as with_the_same_dom Functional_Sequence of X,REAL by MESFUN9C:2;
A137: for n being Nat holds
( F2 . n in Lp_Functions (M,k) & F2 . n in Sq . (N . n) )
proof
let n1 be Nat; ::_thesis: ( F2 . n1 in Lp_Functions (M,k) & F2 . n1 in Sq . (N . n1) )
 F2 . n1 = (F1 . n1) | E0 by MESFUN9C:def_1;
then  abs (F2 . n1) = (abs (F1 . n1)) | E0 by Th13;
then A138: ((abs (F1 . n1)) to_power k) | E0 = (abs (F2 . n1)) to_power k by Th20;
A139: ( F2 . n1 is_measurable_on E0 & dom (F2 . n1) = E0 ) by A135;
 F1 . n1 in Lp_Functions (M,k) by A7;
then  ex FMF being PartFunc of X,REAL st
( F1 . n1 = FMF & ex ND being Element of S st
( M . (ND `) = 0 & dom FMF = ND & FMF is_measurable_on ND & (abs FMF) to_power k is_integrable_on M ) ) ;
then  (abs (F2 . n1)) to_power k is_integrable_on M by A138, MESFUNC6:91;
hence A140: F2 . n1 in Lp_Functions (M,k) by A139, A89; ::_thesis: F2 . n1 in Sq . (N . n1)
reconsider n = n1 as Element of NAT by ORDINAL1:def_12;
set m = N . n;
 F1 . n = Fsq . (N . n) by A6;
then A141: ( F1 . n in Sq . (N . n) & Sq . (N . n) = a.e-eq-class_Lp ((F1 . n),M,k) ) by A3;
reconsider EB = E0 ` as Element of S by MEASURE1:34;
 (F2 . n) | (EB `) = F2 . n by A139, RELAT_1:68;
then  (F2 . n) | (EB `) = (F1 . n) | (EB `) by MESFUN9C:def_1;
then  F2 . n a.e.= F1 . n,M by A89, LPSPACE1:def_10;
hence  F2 . n1 in Sq . (N . n1) by A140, A141, Th36; ::_thesis: verum
end;
A142: dom (lim F2) = dom (F2 . 0) by MESFUNC8:def_9;
then A143: dom (lim F2) = E0 by A135;
A144: for x being Element of X st x in E0 holds
(lim F2) . x = lim (F2 # x)
proof
let x be Element of X; ::_thesis: ( x in E0 implies (lim F2) . x = lim (F2 # x) )
assume  x in E0 ; ::_thesis: (lim F2) . x = lim (F2 # x)
then  ( (lim F2) . x = lim (R_EAL (F2 # x)) & F2 # x is convergent ) by A143, A131, MESFUN7C:14;
hence  (lim F2) . x = lim (F2 # x) by RINFSUP2:14; ::_thesis: verum
end;
now__::_thesis:_for_y_being_set_st_y_in_rng_(lim_F2)_holds_
y_in_REAL
let y be set ; ::_thesis: ( y in rng (lim F2) implies y in REAL )
assume  y in rng (lim F2) ; ::_thesis: y in REAL
then consider x being Element of X such that
A145: ( x in dom (lim F2) & y = (lim F2) . x ) by PARTFUN1:3;
 y = lim (F2 # x) by A145, A143, A144;
hence  y in REAL ; ::_thesis: verum
end;
then  rng (lim F2) c= REAL by TARSKI:def_3;
then reconsider F = lim F2 as PartFunc of X,REAL by A142, RELSET_1:4;
A146: dom (LExtGp | E0) = E /\ E0 by A81, RELAT_1:61;
then A147: dom (LExtGp | E0) = E0 by XBOOLE_1:28, XBOOLE_1:36;
now__::_thesis:_for_y_being_set_st_y_in_rng_(LExtGp_|_E0)_holds_
y_in_REAL
let y be set ; ::_thesis: ( y in rng (LExtGp | E0) implies y in REAL )
assume  y in rng (LExtGp | E0) ; ::_thesis: y in REAL
then consider x being Element of X such that
A148: ( x in dom (LExtGp | E0) & y = (LExtGp | E0) . x ) by PARTFUN1:3;
 y = LExtGp . x by A147, A148, FUNCT_1:49;
hence  y in REAL by A147, A148, A90; ::_thesis: verum
end;
then  rng (LExtGp | E0) c= REAL by TARSKI:def_3;
then reconsider gp = LExtGp | E0 as PartFunc of X,REAL by A146, RELSET_1:4;
A149: for x being Element of X st x in E0 holds
gp . x = lim (Gp # x)
proof
let x be Element of X; ::_thesis: ( x in E0 implies gp . x = lim (Gp # x) )
assume A150: x in E0 ; ::_thesis: gp . x = lim (Gp # x)
then  x in dom LExtGp by A81, XBOOLE_0:def_5;
then  LExtGp . x = lim (ExtGp # x) by MESFUNC8:def_9;
then  gp . x = lim (ExtGp # x) by A150, FUNCT_1:49;
hence  gp . x = lim (Gp # x) by A91, A150; ::_thesis: verum
end;
 LExtGp is nonnegative by A82, SUPINF_2:52;
then  LExtGp is_integrable_on M by A80, A38, A81, Th2;
then  R_EAL gp is_integrable_on M by MESFUNC5:97;
then A151: gp is_integrable_on M by MESFUNC6:def_4;
A152: dom (F2 . 0) = E0 by A135;
then A153: dom F = E0 by MESFUNC8:def_9;
then A154: E0 = dom (abs F) by VALUED_1:def_11;
then A155: E0 = dom ((abs F) to_power k) by MESFUN6C:def_4;
A156: for x being Element of X
for n being Element of NAT st x in E0 holds
(abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) <= (G # x) . n
proof
let x be Element of X; ::_thesis: for n being Element of NAT st x in E0 holds
(abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) <= (G # x) . n
let n be Element of NAT ; ::_thesis: ( x in E0 implies (abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) <= (G # x) . n )
assume A157: x in E0 ; ::_thesis: (abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) <= (G # x) . n
then  x in E by XBOOLE_0:def_5;
then A158: x in dom (G . n) by A17;
 G . n = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . n) by A16;
then  (G . n) . x = ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . n) . x) by A158, VALUED_1:def_1;
then A159: (G . n) . x = (abs ((F1 . 0) . x)) + (((Partial_Sums AbsFMF) . n) . x) by VALUED_1:18;
(G # x) . n = (G . n) . x by SEQFUNC:def_10
.= (abs ((F1 . 0) . x)) + (((Partial_Sums AbsFMF) # x) . n) by A159, SEQFUNC:def_10
.= (abs ((F1 # x) . 0)) + (((Partial_Sums AbsFMF) # x) . n) by SEQFUNC:def_10 ;
then A160: (G # x) . n = (abs ((F1 # x) . 0)) + ((Partial_Sums (abs (FMF # x))) . n) by A108, A157;
 abs ((Partial_Sums (FMF # x)) . n) <= (Partial_Sums (abs (FMF # x))) . n by Lm1;
hence  (abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) <= (G # x) . n by A160, XREAL_1:6; ::_thesis: verum
end;
A161: for x being Element of X
for n being Element of NAT st x in E0 holds
(abs (((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n))) to_power k <= (Gp # x) . n
proof
let x be Element of X; ::_thesis: for n being Element of NAT st x in E0 holds
(abs (((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n))) to_power k <= (Gp # x) . n
let n be Element of NAT ; ::_thesis: ( x in E0 implies (abs (((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n))) to_power k <= (Gp # x) . n )
assume A162: x in E0 ; ::_thesis: (abs (((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n))) to_power k <= (Gp # x) . n
then A163: (Gp # x) . n = ((G # x) . n) to_power k by A96;
A164: (abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) <= (G # x) . n by A156, A162;
 abs (((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n)) <= (abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) by COMPLEX1:56;
then A165: abs (((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n)) <= (G # x) . n by A164, XXREAL_0:2;
 0 <= abs (((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n)) by COMPLEX1:46;
hence  (abs (((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n))) to_power k <= (Gp # x) . n by A163, A165, HOLDER_1:3; ::_thesis: verum
end;
A166: for x being Element of X
for n being Element of NAT st x in E0 holds
(abs ((F2 # x) . n)) to_power k <= (Gp # x) . n
proof
let x be Element of X; ::_thesis: for n being Element of NAT st x in E0 holds
(abs ((F2 # x) . n)) to_power k <= (Gp # x) . n
let n be Element of NAT ; ::_thesis: ( x in E0 implies (abs ((F2 # x) . n)) to_power k <= (Gp # x) . n )
assume A167: x in E0 ; ::_thesis: (abs ((F2 # x) . n)) to_power k <= (Gp # x) . n
then A168: F1 # x = F2 # x by A133;
percases ( n = 0 or n <> 0 ) ;
supposeA169: n = 0 ; ::_thesis: (abs ((F2 # x) . n)) to_power k <= (Gp # x) . n
A170: (Gp # x) . n = ((G # x) . n) to_power k by A167, A96;
A171: (abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) <= (G # x) . n by A156, A167;
 0 <= abs ((Partial_Sums (FMF # x)) . n) by COMPLEX1:46;
then  0 + (abs ((F1 # x) . 0)) <= (abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) by XREAL_1:6;
then A172: abs ((F1 # x) . 0) <= (G # x) . n by A171, XXREAL_0:2;
 0 <= abs ((F1 # x) . 0) by COMPLEX1:46;
hence  (abs ((F2 # x) . n)) to_power k <= (Gp # x) . n by A168, A169, A170, A172, HOLDER_1:3; ::_thesis: verum
end;
suppose n <> 0 ; ::_thesis: (abs ((F2 # x) . n)) to_power k <= (Gp # x) . n
then consider m being Nat such that
A173: n = m + 1 by NAT_1:6;
reconsider m = m as Element of NAT by ORDINAL1:def_12;
 (F1 # x) . (m + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . m) by A116, A167;
then A174: (abs ((F1 # x) . (m + 1))) to_power k <= (Gp # x) . m by A161, A167;
 x in E by A167, XBOOLE_0:def_5;
then A175: ExtGp # x is V169() by A35;
 m <= m + 1 by NAT_1:11;
then A176: (ExtGp # x) . m <= (ExtGp # x) . (m + 1) by A175, RINFSUP2:7;
 ExtGp # x = Gp # x by MESFUN7C:1;
hence  (abs ((F2 # x) . n)) to_power k <= (Gp # x) . n by A168, A173, A174, A176, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A177: for x being Element of X st x in E0 holds
abs (((abs F) to_power k) . x) <= gp . x
proof
let x be Element of X; ::_thesis: ( x in E0 implies abs (((abs F) to_power k) . x) <= gp . x )
assume A178: x in E0 ; ::_thesis: abs (((abs F) to_power k) . x) <= gp . x
then A179: Gp # x is convergent by A91;
deffunc H8( set ) ->  Element of REAL = ((abs (F2 # x)) . $1) to_power k;
consider s being Real_Sequence such that
A180: for n being Element of NAT holds s . n = H8(n) from SEQ_1:sch_1();
A181: gp . x = lim (Gp # x) by A149, A178;
A182: ((abs F) to_power k) . x = ((abs F) . x) to_power k by A155, A178, MESFUN6C:def_4
.= (abs (F . x)) to_power k by A154, A178, VALUED_1:def_11
.= (abs (lim (F2 # x))) to_power k by A178, A144
.= (lim (abs (F2 # x))) to_power k by A131, A178, SEQ_4:14 ;
A183: now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<=_(abs_(F2_#_x))_._n
let n be Element of NAT ; ::_thesis: 0 <= (abs (F2 # x)) . n
 0 <= abs ((F2 # x) . n) by COMPLEX1:46;
hence  0 <= (abs (F2 # x)) . n by VALUED_1:18; ::_thesis: verum
end;
 abs (F2 # x) is convergent by A178, A131, SEQ_4:13;
then A184: ( s is convergent & (lim (abs (F2 # x))) to_power k = lim s ) by A183, A180, HOLDER_1:10;
now__::_thesis:_for_n_being_Element_of_NAT_holds_s_._n_<=_(Gp_#_x)_._n
let n be Element of NAT ; ::_thesis: s . n <= (Gp # x) . n
 (abs ((F2 # x) . n)) to_power k <= (Gp # x) . n by A166, A178;
then  ((abs (F2 # x)) . n) to_power k <= (Gp # x) . n by VALUED_1:18;
hence  s . n <= (Gp # x) . n by A180; ::_thesis: verum
end;
then A185: ((abs F) to_power k) . x <= gp . x by A184, A181, A182, A179, SEQ_2:18;
 0 <= ((abs F) to_power k) . x by MESFUNC6:51;
hence  abs (((abs F) to_power k) . x) <= gp . x by A185, ABSVALUE:def_1; ::_thesis: verum
end;
 R_EAL F is_measurable_on E0 by A135, A152, A131, MESFUN7C:21;
then A186: F is_measurable_on E0 by MESFUNC6:def_1;
then A187: abs F is_measurable_on E0 by A153, MESFUNC6:48;
 dom (abs F) = E0 by A153, VALUED_1:def_11;
then  (abs F) to_power k is_measurable_on E0 by A187, MESFUN6C:29;
then  (abs F) to_power k is_integrable_on M by A147, A151, A155, A177, MESFUNC6:96;
then A188: F in Lp_Functions (M,k) by A89, A153, A186;
A189: for x being Element of X
for n, m being Element of NAT st x in E0 & m <= n holds
|.(((F1 # x) . n) - ((F1 # x) . m)).| to_power k <= (Gp # x) . n
proof
let x be Element of X; ::_thesis: for n, m being Element of NAT st x in E0 & m <= n holds
|.(((F1 # x) . n) - ((F1 # x) . m)).| to_power k <= (Gp # x) . n
let n1, m1 be Element of NAT ; ::_thesis: ( x in E0 & m1 <= n1 implies |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1 )
assume A190: ( x in E0 & m1 <= n1 ) ; ::_thesis: |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1
now__::_thesis:_|.(((F1_#_x)_._n1)_-_((F1_#_x)_._m1)).|_to_power_k_<=_(Gp_#_x)_._n1
percases ( m1 = 0 or m1 <> 0 ) ;
supposeA191: m1 = 0 ; ::_thesis: |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1
now__::_thesis:_|.(((F1_#_x)_._n1)_-_((F1_#_x)_._m1)).|_to_power_k_<=_(Gp_#_x)_._n1
percases ( n1 = 0 or n1 <> 0 ) ;
supposeA192: n1 = 0 ; ::_thesis: |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1
 (G . n1) to_power k is nonnegative by A24;
then  Gp . n1 is nonnegative by A23;
then  0 <= (Gp . n1) . x by MESFUNC6:51;
then  0 <= (Gp # x) . n1 by SEQFUNC:def_10;
hence  |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1 by A191, A192, COMPLEX1:44, POWER:def_2; ::_thesis: verum
end;
suppose n1 <> 0 ; ::_thesis: |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1
then consider n being Nat such that
A193: n1 = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A194: (F1 # x) . (n + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n) by A190, A116;
A195: (abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) <= (G # x) . n by A156, A190;
 0 <= abs ((F1 # x) . 0) by COMPLEX1:46;
then  (abs ((Partial_Sums (FMF # x)) . n)) + 0 <= (abs ((F1 # x) . 0)) + (abs ((Partial_Sums (FMF # x)) . n)) by XREAL_1:6;
then A196: abs ((Partial_Sums (FMF # x)) . n) <= (G # x) . n by A195, XXREAL_0:2;
 0 <= abs ((Partial_Sums (FMF # x)) . n) by COMPLEX1:46;
then A197: |.((Partial_Sums (FMF # x)) . n).| to_power k <= ((G # x) . n) to_power k by A196, HOLDER_1:3;
A198: (Gp # x) . n = ((G # x) . n) to_power k by A190, A96;
 x in E by A190, XBOOLE_0:def_5;
then A199: ExtGp # x is V169() by A35;
 n <= n + 1 by NAT_1:11;
then A200: (ExtGp # x) . n <= (ExtGp # x) . (n + 1) by A199, RINFSUP2:7;
 ExtGp # x = Gp # x by MESFUN7C:1;
hence  |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1 by A191, A193, A200, A197, A198, A194, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1 ; ::_thesis: verum
end;
supposeA201: m1 <> 0 ; ::_thesis: |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1
then consider m being Nat such that
A202: m1 = m + 1 by NAT_1:6;
reconsider m = m as Element of NAT by ORDINAL1:def_12;
 0 < n1 by A190, A201;
then consider n being Nat such that
A203: n1 = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A204: m1 - 1 <= n1 - 1 by A190, XREAL_1:9;
 x in E by A190, XBOOLE_0:def_5;
then A205: x in dom (G . n) by A17;
then A206: x in dom ((G . n) to_power k) by MESFUN6C:def_4;
 (Gp # x) . n = (Gp . n) . x by SEQFUNC:def_10;
then  (Gp # x) . n = ((G . n) to_power k) . x by A23;
then A207: (Gp # x) . n = ((G . n) . x) to_power k by A206, MESFUN6C:def_4;
 G . n = (abs (F1 . 0)) + ((Partial_Sums AbsFMF) . n) by A16;
then (G . n) . x = ((abs (F1 . 0)) . x) + (((Partial_Sums AbsFMF) . n) . x) by A205, VALUED_1:def_1
.= (abs ((F1 . 0) . x)) + (((Partial_Sums AbsFMF) . n) . x) by VALUED_1:18 ;
then A208: (G . n) . x = (abs ((F1 . 0) . x)) + (((Partial_Sums AbsFMF) # x) . n) by SEQFUNC:def_10;
A209: ( (F1 # x) . (n + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . n) & (F1 # x) . (m + 1) = ((F1 # x) . 0) + ((Partial_Sums (FMF # x)) . m) ) by A190, A116;
A210: |.(((Partial_Sums (FMF # x)) . n) - ((Partial_Sums (FMF # x)) . m)).| <= (Partial_Sums (abs (FMF # x))) . n by Th10, A202, A203, A204;
A211: (Partial_Sums (abs (FMF # x))) . n = ((Partial_Sums AbsFMF) # x) . n by A108, A190;
 0 <= abs ((F1 . 0) . x) by COMPLEX1:46;
then  0 + ((Partial_Sums (abs (FMF # x))) . n) <= (abs ((F1 . 0) . x)) + (((Partial_Sums AbsFMF) # x) . n) by A211, XREAL_1:6;
then A212: |.(((F1 # x) . (n + 1)) - ((F1 # x) . (m + 1))).| <= (G . n) . x by A208, A209, A210, XXREAL_0:2;
 0 <= |.(((F1 # x) . (n + 1)) - ((F1 # x) . (m + 1))).| by COMPLEX1:46;
then A213: |.(((F1 # x) . (n + 1)) - ((F1 # x) . (m + 1))).| to_power k <= (Gp # x) . n by A207, A212, HOLDER_1:3;
 x in E by A190, XBOOLE_0:def_5;
then A214: ExtGp # x is V169() by A35;
 n <= n + 1 by NAT_1:11;
then A215: (ExtGp # x) . n <= (ExtGp # x) . (n + 1) by A214, RINFSUP2:7;
 ExtGp # x = Gp # x by MESFUN7C:1;
hence  |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1 by A202, A203, A215, A213, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  |.(((F1 # x) . n1) - ((F1 # x) . m1)).| to_power k <= (Gp # x) . n1 ; ::_thesis: verum
end;
A216: for x being Element of X
for n being Nat st x in E0 holds
|.((F . x) - ((F2 # x) . n)).| to_power k <= gp . x
proof
let x be Element of X; ::_thesis: for n being Nat st x in E0 holds
|.((F . x) - ((F2 # x) . n)).| to_power k <= gp . x
let n1 be Nat; ::_thesis: ( x in E0 implies |.((F . x) - ((F2 # x) . n1)).| to_power k <= gp . x )
assume A217: x in E0 ; ::_thesis: |.((F . x) - ((F2 # x) . n1)).| to_power k <= gp . x
then A218: Gp # x is convergent by A91;
A219: F1 # x = F2 # x by A133, A217;
A220: F2 # x is convergent by A217, A131;
reconsider n = n1 as Element of NAT by ORDINAL1:def_12;
deffunc H8( Element of NAT ) ->  Element of REAL = ((F2 # x) . $1) - ((F2 # x) . n);
consider s0 being Real_Sequence such that
A221: for j being Element of NAT holds s0 . j = H8(j) from SEQ_1:sch_1();
A222: now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_
ex_n1_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n1_<=_m_holds_
abs_((s0_._m)_-_((lim_(F2_#_x))_-_((F2_#_x)_._n)))_<_p
let p be real number ; ::_thesis: ( 0 < p implies ex n1 being Element of NAT st
for m being Element of NAT st n1 <= m holds
abs ((s0 . m) - ((lim (F2 # x)) - ((F2 # x) . n))) < p )
assume  0 < p ; ::_thesis: ex n1 being Element of NAT st
for m being Element of NAT st n1 <= m holds
abs ((s0 . m) - ((lim (F2 # x)) - ((F2 # x) . n))) < p
then consider n1 being Element of NAT  such that
A223: for m being Element of NAT st n1 <= m holds
abs (((F2 # x) . m) - (lim (F2 # x))) < p by A220, SEQ_2:def_7;
take n1 = n1; ::_thesis: for m being Element of NAT st n1 <= m holds
abs ((s0 . m) - ((lim (F2 # x)) - ((F2 # x) . n))) < p
thus  for m being Element of NAT st n1 <= m holds
abs ((s0 . m) - ((lim (F2 # x)) - ((F2 # x) . n))) < p ::_thesis: verum
proof
let m be Element of NAT ; ::_thesis: ( n1 <= m implies abs ((s0 . m) - ((lim (F2 # x)) - ((F2 # x) . n))) < p )
assume A224: n1 <= m ; ::_thesis: abs ((s0 . m) - ((lim (F2 # x)) - ((F2 # x) . n))) < p
 (s0 . m) - ((lim (F2 # x)) - ((F2 # x) . n)) = (((F2 # x) . m) - ((F2 # x) . n)) - ((lim (F2 # x)) - ((F2 # x) . n)) by A221;
then  (s0 . m) - ((lim (F2 # x)) - ((F2 # x) . n)) = ((F2 # x) . m) - (lim (F2 # x)) ;
hence  abs ((s0 . m) - ((lim (F2 # x)) - ((F2 # x) . n))) < p by A224, A223; ::_thesis: verum
end;
end;
then A225: s0 is convergent by SEQ_2:def_6;
then  lim s0 = (lim (F2 # x)) - ((F2 # x) . n) by A222, SEQ_2:def_7;
then A226: lim (abs s0) = abs ((lim (F2 # x)) - ((F2 # x) . n)) by A225, SEQ_4:14;
A227: abs s0 is convergent by A225;
deffunc H9( Element of NAT ) ->  Element of REAL = |.(((F2 # x) . $1) - ((F2 # x) . n)).| to_power k;
consider s being Real_Sequence such that
A228: for j being Element of NAT holds s . j = H9(j) from SEQ_1:sch_1();
A229: for j being Element of NAT st n <= j holds
s . j <= (Gp # x) . j
proof
let j be Element of NAT ; ::_thesis: ( n <= j implies s . j <= (Gp # x) . j )
assume  n <= j ; ::_thesis: s . j <= (Gp # x) . j
then  |.(((F2 # x) . j) - ((F2 # x) . n)).| to_power k <= (Gp # x) . j by A219, A217, A189;
hence  s . j <= (Gp # x) . j by A228; ::_thesis: verum
end;
A230: now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<=_(abs_s0)_._n
let n be Element of NAT ; ::_thesis: 0 <= (abs s0) . n
 0 <= abs (s0 . n) by COMPLEX1:46;
hence  0 <= (abs s0) . n by VALUED_1:18; ::_thesis: verum
end;
now__::_thesis:_for_j_being_Element_of_NAT_holds_s_._j_=_((abs_s0)_._j)_to_power_k
let j be Element of NAT ; ::_thesis: s . j = ((abs s0) . j) to_power k
thus s . j = |.(((F2 # x) . j) - ((F2 # x) . n)).| to_power k by A228
.= (abs (s0 . j)) to_power k by A221
.= ((abs s0) . j) to_power k by VALUED_1:18 ; ::_thesis: verum
end;
then A231: ( s is convergent & lim s = (lim (abs s0)) to_power k ) by A230, A227, HOLDER_1:10;
then A232: ( s ^\ n is convergent & lim (s ^\ n) = lim s ) by SEQ_4:20;
 gp . x = lim (Gp # x) by A149, A217;
then A233: ( (Gp # x) ^\ n is convergent & lim ((Gp # x) ^\ n) = gp . x ) by A218, SEQ_4:20;
 for j being Element of NAT holds (s ^\ n) . j <= ((Gp # x) ^\ n) . j
proof
let j be Element of NAT ; ::_thesis: (s ^\ n) . j <= ((Gp # x) ^\ n) . j
 ( (s ^\ n) . j = s . (n + j) & ((Gp # x) ^\ n) . j = (Gp # x) . (n + j) ) by NAT_1:def_3;
hence  (s ^\ n) . j <= ((Gp # x) ^\ n) . j by A229, NAT_1:11; ::_thesis: verum
end;
then  lim s <= gp . x by A232, A233, SEQ_2:18;
hence  |.((F . x) - ((F2 # x) . n1)).| to_power k <= gp . x by A226, A231, A144, A217; ::_thesis: verum
end;
deffunc H8( Nat) ->  Element of bool [:X,REAL:] = |.(F - (F2 . $1)).| to_power k;
consider FP being Functional_Sequence of X,REAL such that
A234: for n being Nat holds FP . n = H8(n) from SEQFUNC:sch_1();
A235: for n being Nat holds dom (FP . n) = E0
proof
let n1 be Nat; ::_thesis: dom (FP . n1) = E0
reconsider n = n1 as Element of NAT by ORDINAL1:def_12;
A236: dom (F2 . n) = E0 by A135;
 dom (FP . n1) = dom ((abs (F - (F2 . n))) to_power k) by A234;
then  dom (FP . n1) = dom (abs (F - (F2 . n))) by MESFUN6C:def_4;
then  dom (FP . n1) = dom (F - (F2 . n)) by VALUED_1:def_11;
then  dom (FP . n1) = E0 /\ E0 by A236, A143, VALUED_1:12;
hence  dom (FP . n1) = E0 ; ::_thesis: verum
end;
then A237: E0 = dom (FP . 0) ;
then A238: dom (lim FP) = E0 by MESFUNC8:def_9;
 for n, m being Nat holds dom (FP . n) = dom (FP . m)
proof
let n, m be Nat; ::_thesis: dom (FP . n) = dom (FP . m)
thus  dom (FP . n) = E0 by A235
.= dom (FP . m) by A235 ; ::_thesis: verum
end;
then reconsider FP = FP as with_the_same_dom Functional_Sequence of X,REAL by MESFUNC8:def_2;
A239: for n being Nat holds FP . n is_measurable_on E0
proof
let n1 be Nat; ::_thesis: FP . n1 is_measurable_on E0
reconsider n = n1 as Element of NAT by ORDINAL1:def_12;
 dom (F2 . n) = E0 by A135;
then A240: dom (F - (F2 . n)) = E0 /\ E0 by A143, VALUED_1:12;
 ( F2 . n is_measurable_on E0 & dom (F2 . n) = E0 ) by A135;
then  F - (F2 . n) is_measurable_on E0 by A186, MESFUNC6:29;
then A241: abs (F - (F2 . n)) is_measurable_on E0 by A240, MESFUNC6:48;
 dom (abs (F - (F2 . n))) = E0 by A240, VALUED_1:def_11;
then  (abs (F - (F2 . n))) to_power k is_measurable_on E0 by A241, MESFUN6C:29;
hence  FP . n1 is_measurable_on E0 by A234; ::_thesis: verum
end;
 for x being Element of X
for n being Nat st x in E0 holds
|.(FP . n).| . x <= gp . x
proof
let x be Element of X; ::_thesis: for n being Nat st x in E0 holds
|.(FP . n).| . x <= gp . x
let n1 be Nat; ::_thesis: ( x in E0 implies |.(FP . n1).| . x <= gp . x )
reconsider n = n1 as Element of NAT by ORDINAL1:def_12;
assume A242: x in E0 ; ::_thesis: |.(FP . n1).| . x <= gp . x
then A243: x in dom (FP . n) by A235;
then  x in dom (|.(F - (F2 . n)).| to_power k) by A234;
then  x in dom |.(F - (F2 . n)).| by MESFUN6C:def_4;
then A244: x in dom (F - (F2 . n)) by VALUED_1:def_11;
A245: FP . n1 = |.(F - (F2 . n1)).| to_power k by A234;
A246: 0 <= |.((F . x) - ((F2 . n1) . x)).| to_power k by Th4, COMPLEX1:46;
|.(FP . n).| . x = |.((FP . n) . x).| by VALUED_1:18
.= |.((|.(F - (F2 . n1)).| . x) to_power k).| by A243, A245, MESFUN6C:def_4
.= |.(|.((F - (F2 . n1)) . x).| to_power k).| by VALUED_1:18
.= |.(|.((F . x) - ((F2 . n1) . x)).| to_power k).| by A244, VALUED_1:13
.= |.((F . x) - ((F2 . n1) . x)).| to_power k by A246, ABSVALUE:def_1
.= |.((F . x) - ((F2 # x) . n)).| to_power k by SEQFUNC:def_10 ;
hence  |.(FP . n1).| . x <= gp . x by A216, A242; ::_thesis: verum
end;
then consider Ip being Real_Sequence such that
A247: ( ( for n being Nat holds Ip . n = Integral (M,(FP . n)) ) & ( ( for x being Element of X st x in E0 holds
FP # x is convergent ) implies ( Ip is convergent & lim Ip = Integral (M,(lim FP)) ) ) ) by A147, A151, A237, A239, MESFUN9C:48;
A248: for x being Element of X st x in E0 holds
( FP # x is convergent & lim (FP # x) = 0 )
proof
let x be Element of X; ::_thesis: ( x in E0 implies ( FP # x is convergent & lim (FP # x) = 0 ) )
assume A249: x in E0 ; ::_thesis: ( FP # x is convergent & lim (FP # x) = 0 )
A250: for n being Element of NAT holds (FP # x) . n = |.((lim (F2 # x)) - ((F2 # x) . n)).| to_power k
proof
let n be Element of NAT ; ::_thesis: (FP # x) . n = |.((lim (F2 # x)) - ((F2 # x) . n)).| to_power k
 x in dom (FP . n) by A249, A235;
then A251: x in dom (|.(F - (F2 . n)).| to_power k) by A234;
then  x in dom |.(F - (F2 . n)).| by MESFUN6C:def_4;
then A252: x in dom (F - (F2 . n)) by VALUED_1:def_11;
thus (FP # x) . n = (FP . n) . x by SEQFUNC:def_10
.= (|.(F - (F2 . n)).| to_power k) . x by A234
.= (|.(F - (F2 . n)).| . x) to_power k by A251, MESFUN6C:def_4
.= |.((F - (F2 . n)) . x).| to_power k by VALUED_1:18
.= |.((F . x) - ((F2 . n) . x)).| to_power k by A252, VALUED_1:13
.= |.((lim (F2 # x)) - ((F2 . n) . x)).| to_power k by A144, A249
.= |.((lim (F2 # x)) - ((F2 # x) . n)).| to_power k by SEQFUNC:def_10 ; ::_thesis: verum
end;
 F2 # x is convergent by A249, A131;
hence  ( FP # x is convergent & lim (FP # x) = 0 ) by A250, Th11; ::_thesis: verum
end;
A253: for x being Element of X st x in dom (lim FP) holds
0 = (lim FP) . x
proof
let x be Element of X; ::_thesis: ( x in dom (lim FP) implies 0 = (lim FP) . x )
assume A254: x in dom (lim FP) ; ::_thesis: 0 = (lim FP) . x
then A255: ( lim (FP # x) = 0 & FP # x is convergent ) by A248, A238;
 (lim FP) . x = lim (R_EAL (FP # x)) by A254, MESFUN7C:14;
hence  0 = (lim FP) . x by A255, RINFSUP2:14; ::_thesis: verum
end;
 a.e-eq-class_Lp (F,M,k) in CosetSet (M,k) by A188;
then reconsider Sq0 = a.e-eq-class_Lp (F,M,k) as Point of (Lp-Space (M,k)) by Def11;
A256: for n being Element of NAT holds Ip . n = ||.(Sq0 - (Sq . (N . n))).|| to_power k
proof
let n be Element of NAT ; ::_thesis: Ip . n = ||.(Sq0 - (Sq . (N . n))).|| to_power k
set m = N . n;
reconsider n1 = n as Nat ;
A257: FP . n = (abs (F - (F2 . n1))) to_power k by A234;
A258: ( F in Lp_Functions (M,k) & F in Sq0 ) by A188, Th38;
 ( F2 . n1 in Lp_Functions (M,k) & F2 . n1 in Sq . (N . n) ) by A137;
then  (- 1) (#) (F2 . n1) in (- 1) * (Sq . (N . n)) by Th54;
then  F - (F2 . n1) in Sq0 + ((- 1) * (Sq . (N . n))) by Th54, A258;
then  F - (F2 . n1) in Sq0 - (Sq . (N . n)) by RLVECT_1:16;
then consider r being Real such that
A259: ( 0 <= r & r = Integral (M,((abs (F - (F2 . n1))) to_power k)) & ||.(Sq0 - (Sq . (N . n))).|| = r to_power (1 / k) ) by Th53;
||.(Sq0 - (Sq . (N . n))).|| to_power k = r to_power ((1 / k) * k) by A259, HOLDER_1:2
.= r to_power 1 by XCMPLX_1:106
.= r by POWER:25 ;
hence  Ip . n = ||.(Sq0 - (Sq . (N . n))).|| to_power k by A259, A257, A247; ::_thesis: verum
end;
deffunc H9( Element of NAT ) ->  Element of REAL = ||.(Sq0 - (Sq . (N . $1))).||;
consider Iq being Real_Sequence such that
A260: for n being Element of NAT holds Iq . n = H9(n) from SEQ_1:sch_1();
A261: now__::_thesis:_for_n_being_Nat_holds_Iq_._n_=_||.(Sq0_-_(Sq_._(N_._n))).||
let n be Nat; ::_thesis: Iq . n = ||.(Sq0 - (Sq . (N . n))).||
reconsider n0 = n as Element of NAT by ORDINAL1:def_12;
 Iq . n = ||.(Sq0 - (Sq . (N . n0))).|| by A260;
hence  Iq . n = ||.(Sq0 - (Sq . (N . n))).|| ; ::_thesis: verum
end;
 ( Iq is convergent & lim Iq = 0 )
proof
A262: for n being Element of NAT holds Ip . n >= 0
proof
let n be Element of NAT ; ::_thesis: Ip . n >= 0
 ||.(Sq0 - (Sq . (N . n))).|| to_power k >= 0 by Th4;
hence  Ip . n >= 0 by A256; ::_thesis: verum
end;
A263: for n being Element of NAT holds Iq . n = (Ip . n) to_power (1 / k)
proof
let n be Element of NAT ; ::_thesis: Iq . n = (Ip . n) to_power (1 / k)
thus (Ip . n) to_power (1 / k) = (||.(Sq0 - (Sq . (N . n))).|| to_power k) to_power (1 / k) by A256
.= ||.(Sq0 - (Sq . (N . n))).|| to_power (k * (1 / k)) by HOLDER_1:2
.= ||.(Sq0 - (Sq . (N . n))).|| to_power 1 by XCMPLX_1:106
.= ||.(Sq0 - (Sq . (N . n))).|| by POWER:25
.= Iq . n by A261 ; ::_thesis: verum
end;
hence  Iq is convergent by A262, A248, A247, HOLDER_1:10; ::_thesis: lim Iq = 0
 lim Iq = (lim Ip) to_power (1 / k) by A248, A247, A262, A263, HOLDER_1:10;
then  lim Iq = 0 to_power (1 / k) by A248, A247, A253, A238, LPSPACE1:22;
hence  lim Iq = 0 by POWER:def_2; ::_thesis: verum
end;
hence  Sq is convergent by A2, A261, Lm7; ::_thesis: verum
end;

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let k be geq_than_1 Real;
cluster Lp-Space (M,k) ->  non empty complete ;
coherence
 Lp-Space (M,k) is complete
proof
 for Sq being sequence of (Lp-Space (M,k)) st Sq is Cauchy_sequence_by_Norm holds
Sq is convergent by Th70;
hence  Lp-Space (M,k) is complete by LOPBAN_1:def_15; ::_thesis: verum
end;
end;

begin

Lm8:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st f in L1_Functions M holds
( f is_integrable_on M & ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) )
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st f in L1_Functions M holds
( f is_integrable_on M & ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) )
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st f in L1_Functions M holds
( f is_integrable_on M & ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) )
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL st f in L1_Functions M holds
( f is_integrable_on M & ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) )
let f be PartFunc of X,REAL; ::_thesis: ( f in L1_Functions M implies ( f is_integrable_on M & ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) ) )
assume  f in L1_Functions M ; ::_thesis: ( f is_integrable_on M & ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) )
then  ex f2 being PartFunc of X,REAL st
( f = f2 & ex E being Element of S st
( M . E = 0 & dom f2 = E ` & f2 is_integrable_on M ) ) ;
then consider D being Element of S such that
A1: ( M . D = 0 & dom f = D ` & f is_integrable_on M ) ;
thus  f is_integrable_on M by A1; ::_thesis: ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E )
reconsider E = D ` as Element of S by MEASURE1:34;
take  E ; ::_thesis: ( M . (E `) = 0 & E = dom f & f is_measurable_on E )
thus  ( M . (E `) = 0 & dom f = E ) by A1; ::_thesis: f is_measurable_on E
 R_EAL f is_integrable_on M by A1, MESFUNC6:def_4;
then  ex B being Element of S st
( B = dom (R_EAL f) & R_EAL f is_measurable_on B ) by MESFUNC5:def_17;
hence  f is_measurable_on E by A1, MESFUNC6:def_1; ::_thesis: verum
end;

Lm9:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
(abs f) to_power k is_integrable_on M
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
(abs f) to_power k is_integrable_on M
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
(abs f) to_power k is_integrable_on M
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions (M,k) holds
(abs f) to_power k is_integrable_on M
let f be PartFunc of X,REAL; ::_thesis: for k being positive Real st f in Lp_Functions (M,k) holds
(abs f) to_power k is_integrable_on M
let k be positive Real; ::_thesis: ( f in Lp_Functions (M,k) implies (abs f) to_power k is_integrable_on M )
assume  f in Lp_Functions (M,k) ; ::_thesis: (abs f) to_power k is_integrable_on M
then  ex f2 being PartFunc of X,REAL st
( f = f2 & ex E being Element of S st
( M . (E `) = 0 & dom f2 = E & f2 is_measurable_on E & (abs f2) to_power k is_integrable_on M ) ) ;
hence  (abs f) to_power k is_integrable_on M ; ::_thesis: verum
end;

Lm10:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp (f,M,1) c= a.e-eq-class (f,M)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp (f,M,1) c= a.e-eq-class (f,M)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp (f,M,1) c= a.e-eq-class (f,M)
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp (f,M,1) c= a.e-eq-class (f,M)
let f be PartFunc of X,REAL; ::_thesis: ( ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) implies a.e-eq-class_Lp (f,M,1) c= a.e-eq-class (f,M) )
assume A1: ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) ; ::_thesis: a.e-eq-class_Lp (f,M,1) c= a.e-eq-class (f,M)
now__::_thesis:_for_x_being_set_st_x_in_a.e-eq-class_Lp_(f,M,1)_holds_
x_in_a.e-eq-class_(f,M)
let x be set ; ::_thesis: ( x in a.e-eq-class_Lp (f,M,1) implies x in a.e-eq-class (f,M) )
assume  x in a.e-eq-class_Lp (f,M,1) ; ::_thesis: x in a.e-eq-class (f,M)
then consider h being PartFunc of X,REAL such that
A2: ( x = h & h in Lp_Functions (M,1) & f a.e.= h,M ) ;
A3: ex g being PartFunc of X,REAL st
( h = g & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is_measurable_on E & (abs g) to_power 1 is_integrable_on M ) ) by A2;
then consider Eh being Element of S such that
A4: ( M . (Eh `) = 0 & dom h = Eh & h is_measurable_on Eh & (abs h) to_power 1 is_integrable_on M ) ;
A5: dom ((abs h) to_power 1) = dom (abs h) by MESFUN6C:def_4;
 for x being Element of X st x in dom ((abs h) to_power 1) holds
((abs h) to_power 1) . x = (abs h) . x
proof
let x be Element of X; ::_thesis: ( x in dom ((abs h) to_power 1) implies ((abs h) to_power 1) . x = (abs h) . x )
assume  x in dom ((abs h) to_power 1) ; ::_thesis: ((abs h) to_power 1) . x = (abs h) . x
then  ((abs h) to_power 1) . x = ((abs h) . x) to_power 1 by MESFUN6C:def_4;
hence  ((abs h) to_power 1) . x = (abs h) . x by POWER:25; ::_thesis: verum
end;
then  (abs h) to_power 1 = abs h by A5, PARTFUN1:5;
then A6: h is_integrable_on M by A3, MESFUNC6:94;
reconsider ND = Eh ` as Element of S by MEASURE1:34;
 ( M . ND = 0 & dom h = ND ` ) by A4;
then A7: h in L1_Functions M by A6;
 ex E being Element of S st
( M . E = 0 & dom f = E ` & f is_integrable_on M )
proof
consider Ef being Element of S such that
A8: ( M . (Ef `) = 0 & Ef = dom f & f is_measurable_on Ef ) by A1;
reconsider E = Ef ` as Element of S by MEASURE1:34;
take  E ; ::_thesis: ( M . E = 0 & dom f = E ` & f is_integrable_on M )
consider EE being Element of S such that
A9: ( M . EE = 0 & f | (EE `) = h | (EE `) ) by A2, LPSPACE1:def_10;
reconsider E1 = ND \/ EE as Element of S ;
 ( ND is measure_zero of M & EE is measure_zero of M ) by A4, A9, MEASURE1:def_7;
then  E1 is measure_zero of M by MEASURE1:37;
then A10: M . E1 = 0 by MEASURE1:def_7;
 EE c= E1 by XBOOLE_1:7;
then  E1 ` c= EE ` by SUBSET_1:12;
then A11: ( f | (E1 `) = (f | (EE `)) | (E1 `) & h | (E1 `) = (h | (EE `)) | (E1 `) ) by FUNCT_1:51;
A12: ( dom (max+ (R_EAL f)) = Ef & dom (max- (R_EAL f)) = Ef ) by A8, MESFUNC2:def_2, MESFUNC2:def_3;
A13: ( Ef = dom (R_EAL f) & R_EAL f is_measurable_on Ef ) by A8, MESFUNC6:def_1;
then A14: ( max+ (R_EAL f) is_measurable_on Ef & max- (R_EAL f) is_measurable_on Ef ) by MESFUNC2:25, MESFUNC2:26;
 ( ( for x being Element of X holds 0. <= (max+ (R_EAL f)) . x ) & ( for x being Element of X holds 0. <= (max- (R_EAL f)) . x ) ) by MESFUNC2:12, MESFUNC2:13;
then A15: ( max+ (R_EAL f) is nonnegative & max- (R_EAL f) is nonnegative ) by SUPINF_2:39;
A16: Ef = (Ef /\ E1) \/ (Ef \ E1) by XBOOLE_1:51;
reconsider E0 = Ef /\ E1 as Element of S ;
A17: Ef \ E1 = Ef /\ (E1 `) by SUBSET_1:13;
reconsider E2 = Ef \ E1 as Element of S ;
 ( max+ (R_EAL f) = (max+ (R_EAL f)) | (dom (max+ (R_EAL f))) & max- (R_EAL f) = (max- (R_EAL f)) | (dom (max- (R_EAL f))) ) by RELAT_1:69;
then A18: ( integral+ (M,(max+ (R_EAL f))) = (integral+ (M,((max+ (R_EAL f)) | E0))) + (integral+ (M,((max+ (R_EAL f)) | E2))) & integral+ (M,(max- (R_EAL f))) = (integral+ (M,((max- (R_EAL f)) | E0))) + (integral+ (M,((max- (R_EAL f)) | E2))) ) by A12, A15, A16, A14, MESFUNC5:81, XBOOLE_1:89;
A19: ( integral+ (M,((max+ (R_EAL f)) | E0)) >= 0 & integral+ (M,((max- (R_EAL f)) | E0)) >= 0 ) by A15, A14, A12, MESFUNC5:80;
 ( integral+ (M,((max+ (R_EAL f)) | E1)) = 0 & integral+ (M,((max- (R_EAL f)) | E1)) = 0 ) by A10, A12, A15, A14, MESFUNC5:82;
then  ( integral+ (M,((max+ (R_EAL f)) | E0)) = 0 & integral+ (M,((max- (R_EAL f)) | E0)) = 0 ) by A19, A12, A15, A14, MESFUNC5:83, XBOOLE_1:17;
then A20: ( integral+ (M,(max+ (R_EAL f))) = integral+ (M,((max+ (R_EAL f)) | E2)) & integral+ (M,(max- (R_EAL f))) = integral+ (M,((max- (R_EAL f)) | E2)) ) by A18, XXREAL_3:4;
A21: E2 c= E1 ` by A17, XBOOLE_1:17;
then  f | E2 = (h | (E1 `)) | E2 by A9, A11, FUNCT_1:51;
then A22: (R_EAL f) | E2 = (R_EAL h) | E2 by A21, FUNCT_1:51;
A23: ( (max+ (R_EAL f)) | E2 = max+ ((R_EAL f) | E2) & (max+ (R_EAL h)) | E2 = max+ ((R_EAL h) | E2) & (max- (R_EAL f)) | E2 = max- ((R_EAL f) | E2) & (max- (R_EAL h)) | E2 = max- ((R_EAL h) | E2) ) by MESFUNC5:28;
A24: R_EAL h is_integrable_on M by A6, MESFUNC6:def_4;
then A25: ( integral+ (M,(max+ (R_EAL h))) < +infty & integral+ (M,(max- (R_EAL h))) < +infty ) by MESFUNC5:def_17;
 ( integral+ (M,(max+ ((R_EAL h) | E2))) <= integral+ (M,(max+ (R_EAL h))) & integral+ (M,(max- ((R_EAL h) | E2))) <= integral+ (M,(max- (R_EAL h))) ) by A24, MESFUNC5:97;
then  ( integral+ (M,(max+ (R_EAL f))) < +infty & integral+ (M,(max- (R_EAL f))) < +infty ) by A20, A25, A23, A22, XXREAL_0:2;
then  R_EAL f is_integrable_on M by A13, MESFUNC5:def_17;
hence  ( M . E = 0 & dom f = E ` & f is_integrable_on M ) by A8, MESFUNC6:def_4; ::_thesis: verum
end;
then  f in L1_Functions M ;
hence  x in a.e-eq-class (f,M) by A2, A7; ::_thesis: verum
end;
hence  a.e-eq-class_Lp (f,M,1) c= a.e-eq-class (f,M) by TARSKI:def_3; ::_thesis: verum
end;

Lm11:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL holds a.e-eq-class (f,M) c= a.e-eq-class_Lp (f,M,1)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL holds a.e-eq-class (f,M) c= a.e-eq-class_Lp (f,M,1)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL holds a.e-eq-class (f,M) c= a.e-eq-class_Lp (f,M,1)
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL holds a.e-eq-class (f,M) c= a.e-eq-class_Lp (f,M,1)
let f be PartFunc of X,REAL; ::_thesis: a.e-eq-class (f,M) c= a.e-eq-class_Lp (f,M,1)
now__::_thesis:_for_x_being_set_st_x_in_a.e-eq-class_(f,M)_holds_
x_in_a.e-eq-class_Lp_(f,M,1)
let x be set ; ::_thesis: ( x in a.e-eq-class (f,M) implies x in a.e-eq-class_Lp (f,M,1) )
assume  x in a.e-eq-class (f,M) ; ::_thesis: x in a.e-eq-class_Lp (f,M,1)
then consider g being PartFunc of X,REAL such that
A1: ( x = g & g in L1_Functions M & f in L1_Functions M & f a.e.= g,M ) ;
A2: ex h being PartFunc of X,REAL st
( g = h & ex D being Element of S st
( M . D = 0 & dom h = D ` & h is_integrable_on M ) ) by A1;
then  R_EAL g is_integrable_on M by MESFUNC6:def_4;
then consider A being Element of S such that
A3: ( A = dom (R_EAL g) & R_EAL g is_measurable_on A ) by MESFUNC5:def_17;
A4: ( A = dom g & g is_measurable_on A ) by A3, MESFUNC6:def_1;
A5: M . (A `) = 0 by A2, A3;
 (abs g) to_power 1 = abs g by Th8;
then  (abs g) to_power 1 is_integrable_on M by A2, A4, MESFUNC6:94;
then  g in  {  p where p is PartFunc of X,REAL : ex Ep being Element of S st
( M . (Ep `) = 0 & dom p = Ep & p is_measurable_on Ep & (abs p) to_power 1 is_integrable_on M )  }  by A4, A5;
hence  x in a.e-eq-class_Lp (f,M,1) by A1; ::_thesis: verum
end;
hence  a.e-eq-class (f,M) c= a.e-eq-class_Lp (f,M,1) by TARSKI:def_3; ::_thesis: verum
end;

Lm12:  for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp (f,M,1) = a.e-eq-class (f,M)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp (f,M,1) = a.e-eq-class (f,M)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp (f,M,1) = a.e-eq-class (f,M)
let M be sigma_Measure of S; ::_thesis: for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp (f,M,1) = a.e-eq-class (f,M)
let f be PartFunc of X,REAL; ::_thesis: ( ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) implies a.e-eq-class_Lp (f,M,1) = a.e-eq-class (f,M) )
assume  ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is_measurable_on E ) ; ::_thesis: a.e-eq-class_Lp (f,M,1) = a.e-eq-class (f,M)
then A1: a.e-eq-class_Lp (f,M,1) c= a.e-eq-class (f,M) by Lm10;
 a.e-eq-class (f,M) c= a.e-eq-class_Lp (f,M,1) by Lm11;
hence  a.e-eq-class_Lp (f,M,1) = a.e-eq-class (f,M) by A1, XBOOLE_0:def_10; ::_thesis: verum
end;

theorem Th71: :: LPSPACE2:71
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds CosetSet M = CosetSet (M,1)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S holds CosetSet M = CosetSet (M,1)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S holds CosetSet M = CosetSet (M,1)
let M be sigma_Measure of S; ::_thesis: CosetSet M = CosetSet (M,1)
now__::_thesis:_for_x_being_set_st_x_in_CosetSet_M_holds_
x_in_CosetSet_(M,1)
let x be set ; ::_thesis: ( x in CosetSet M implies x in CosetSet (M,1) )
assume  x in CosetSet M ; ::_thesis: x in CosetSet (M,1)
then consider g being PartFunc of X,REAL such that
A1: ( x = a.e-eq-class (g,M) & g in L1_Functions M ) ;
A2: ( g is_integrable_on M & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is_measurable_on E ) ) by A1, Lm8;
then A3: x = a.e-eq-class_Lp (g,M,1) by A1, Lm12;
 (abs g) to_power 1 = abs g by Th8;
then  (abs g) to_power 1 is_integrable_on M by A2, MESFUNC6:94;
then  g in Lp_Functions (M,1) by A2;
hence  x in CosetSet (M,1) by A3; ::_thesis: verum
end;
then A4: CosetSet M c= CosetSet (M,1) by TARSKI:def_3;
now__::_thesis:_for_x_being_set_st_x_in_CosetSet_(M,1)_holds_
x_in_CosetSet_M
let x be set ; ::_thesis: ( x in CosetSet (M,1) implies x in CosetSet M )
assume  x in CosetSet (M,1) ; ::_thesis: x in CosetSet M
then consider g being PartFunc of X,REAL such that
A5: ( x = a.e-eq-class_Lp (g,M,1) & g in Lp_Functions (M,1) ) ;
consider E being Element of S such that
A6: ( M . (E `) = 0 & dom g = E & g is_measurable_on E ) by A5, Th35;
A7: x = a.e-eq-class (g,M) by A5, A6, Lm12;
reconsider D = E ` as Element of S by MEASURE1:34;
A8: ( M . D = 0 & dom g = D ` ) by A6;
 (abs g) to_power 1 is_integrable_on M by A5, Lm9;
then  abs g is_integrable_on M by Th8;
then  g is_integrable_on M by A6, MESFUNC6:94;
then  g in L1_Functions M by A8;
hence  x in CosetSet M by A7; ::_thesis: verum
end;
then  CosetSet (M,1) c= CosetSet M by TARSKI:def_3;
hence  CosetSet M = CosetSet (M,1) by A4, XBOOLE_0:def_10; ::_thesis: verum
end;

theorem Th72: :: LPSPACE2:72
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds addCoset M = addCoset (M,1)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S holds addCoset M = addCoset (M,1)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S holds addCoset M = addCoset (M,1)
let M be sigma_Measure of S; ::_thesis: addCoset M = addCoset (M,1)
A1: CosetSet M = CosetSet (M,1) by Th71;
now__::_thesis:_for_A,_B_being_Element_of_CosetSet_M_holds_(addCoset_M)_._(A,B)_=_(addCoset_(M,1))_._(A,B)
let A, B be Element of CosetSet M; ::_thesis: (addCoset M) . (A,B) = (addCoset (M,1)) . (A,B)
 A in  {  (a.e-eq-class (f,M)) where f is PartFunc of X,REAL : f in L1_Functions M  }  ;
then consider a being PartFunc of X,REAL such that
A2: ( A = a.e-eq-class (a,M) & a in L1_Functions M ) ;
 B in  {  (a.e-eq-class (f,M)) where f is PartFunc of X,REAL : f in L1_Functions M  }  ;
then consider b being PartFunc of X,REAL such that
A3: ( B = a.e-eq-class (b,M) & b in L1_Functions M ) ;
A4: ( A is Element of CosetSet (M,1) & B is Element of CosetSet (M,1) ) by Th71;
A5: ( a in a.e-eq-class (a,M) & b in a.e-eq-class (b,M) ) by A2, A3, LPSPACE1:38;
then A6: (addCoset M) . (A,B) = a.e-eq-class ((a + b),M) by A2, A3, LPSPACE1:def_15;
 a + b in L1_Functions M by A2, A3, LPSPACE1:23;
then  ex E being Element of S st
( M . (E `) = 0 & E = dom (a + b) & a + b is_measurable_on E ) by Lm8;
then  (addCoset M) . (A,B) = a.e-eq-class_Lp ((a + b),M,1) by A6, Lm12;
hence  (addCoset M) . (A,B) = (addCoset (M,1)) . (A,B) by A4, A5, A2, A3, Def8; ::_thesis: verum
end;
hence  addCoset M = addCoset (M,1) by A1, BINOP_1:2; ::_thesis: verum
end;

theorem Th73: :: LPSPACE2:73
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds zeroCoset M = zeroCoset (M,1)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S holds zeroCoset M = zeroCoset (M,1)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S holds zeroCoset M = zeroCoset (M,1)
let M be sigma_Measure of S; ::_thesis: zeroCoset M = zeroCoset (M,1)
reconsider z = zeroCoset (M,1) as Element of CosetSet M by Th71;
 X --> 0 in Lp_Functions (M,1) by Th23;
then  ex E being Element of S st
( M . (E `) = 0 & dom (X --> 0) = E & X --> 0 is_measurable_on E ) by Th35;
then A1: z = a.e-eq-class ((X --> 0),M) by Lm12;
 X --> 0 in L1_Functions M by Th56;
hence  zeroCoset M = zeroCoset (M,1) by A1, LPSPACE1:def_16; ::_thesis: verum
end;

theorem Th74: :: LPSPACE2:74
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds lmultCoset M = lmultCoset (M,1)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S holds lmultCoset M = lmultCoset (M,1)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S holds lmultCoset M = lmultCoset (M,1)
let M be sigma_Measure of S; ::_thesis: lmultCoset M = lmultCoset (M,1)
A1: CosetSet M = CosetSet (M,1) by Th71;
now__::_thesis:_for_z_being_Element_of_REAL_
for_A_being_Element_of_CosetSet_M_holds_(lmultCoset_M)_._(z,A)_=_(lmultCoset_(M,1))_._(z,A)
let z be Element of REAL ; ::_thesis: for A being Element of CosetSet M holds (lmultCoset M) . (z,A) = (lmultCoset (M,1)) . (z,A)
let A be Element of CosetSet M; ::_thesis: (lmultCoset M) . (z,A) = (lmultCoset (M,1)) . (z,A)
 A in  {  (a.e-eq-class (f,M)) where f is PartFunc of X,REAL : f in L1_Functions M  }  ;
then consider a being PartFunc of X,REAL such that
A2: ( A = a.e-eq-class (a,M) & a in L1_Functions M ) ;
A3: A is Element of CosetSet (M,1) by Th71;
A4: a in A by A2, LPSPACE1:38;
then A5: (lmultCoset M) . (z,A) = a.e-eq-class ((z (#) a),M) by LPSPACE1:def_17;
 z (#) a in L1_Functions M by A2, LPSPACE1:24;
then  ex E being Element of S st
( M . (E `) = 0 & E = dom (z (#) a) & z (#) a is_measurable_on E ) by Lm8;
then  (lmultCoset M) . (z,A) = a.e-eq-class_Lp ((z (#) a),M,1) by A5, Lm12;
hence  (lmultCoset M) . (z,A) = (lmultCoset (M,1)) . (z,A) by A3, A4, Def10; ::_thesis: verum
end;
hence  lmultCoset M = lmultCoset (M,1) by A1, BINOP_1:2; ::_thesis: verum
end;

theorem Th75: :: LPSPACE2:75
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds Pre-L-Space M = Pre-Lp-Space (M,1)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S holds Pre-L-Space M = Pre-Lp-Space (M,1)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S holds Pre-L-Space M = Pre-Lp-Space (M,1)
let M be sigma_Measure of S; ::_thesis: Pre-L-Space M = Pre-Lp-Space (M,1)
A1: ( the carrier of (Pre-L-Space M) = CosetSet M & the addF of (Pre-L-Space M) = addCoset M & 0. (Pre-L-Space M) = zeroCoset M & the Mult of (Pre-L-Space M) = lmultCoset M ) by LPSPACE1:def_18;
 ( CosetSet M = CosetSet (M,1) & addCoset M = addCoset (M,1) & zeroCoset M = zeroCoset (M,1) & lmultCoset M = lmultCoset (M,1) ) by Th71, Th72, Th73, Th74;
hence  Pre-L-Space M = Pre-Lp-Space (M,1) by A1, Def11; ::_thesis: verum
end;

theorem Th76: :: LPSPACE2:76
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds L-1-Norm M = Lp-Norm (M,1)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S holds L-1-Norm M = Lp-Norm (M,1)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S holds L-1-Norm M = Lp-Norm (M,1)
let M be sigma_Measure of S; ::_thesis: L-1-Norm M = Lp-Norm (M,1)
A1: the carrier of (Pre-L-Space M) = the carrier of (Pre-Lp-Space (M,1)) by Th75;
now__::_thesis:_for_x_being_Element_of_the_carrier_of_(Pre-L-Space_M)_holds_(L-1-Norm_M)_._x_=_(Lp-Norm_(M,1))_._x
let x be Element of the carrier of (Pre-L-Space M); ::_thesis: (L-1-Norm M) . x = (Lp-Norm (M,1)) . x
 x in the carrier of (Pre-L-Space M) ;
then  x in CosetSet M by LPSPACE1:def_18;
then consider g being PartFunc of X,REAL such that
A2: ( x = a.e-eq-class (g,M) & g in L1_Functions M ) ;
consider a being PartFunc of X,REAL such that
A3: ( a in x & (L-1-Norm M) . x = Integral (M,(abs a)) ) by LPSPACE1:def_19;
A4: ex p being PartFunc of X,REAL st
( a = p & p in L1_Functions M & g in L1_Functions M & g a.e.= p,M ) by A2, A3;
consider b being PartFunc of X,REAL such that
A5: ( b in x & ex r being Real st
( r = Integral (M,((abs b) to_power 1)) & (Lp-Norm (M,1)) . x = r to_power (1 / 1) ) ) by A1, Def12;
A6: ex q being PartFunc of X,REAL st
( b = q & q in L1_Functions M & g in L1_Functions M & g a.e.= q,M ) by A2, A5;
 a a.e.= g,M by A4, LPSPACE1:29;
then  a a.e.= b,M by A6, LPSPACE1:30;
then A7: Integral (M,(abs a)) = Integral (M,(abs b)) by A2, A3, A5, LPSPACE1:45;
 (abs b) to_power 1 = abs b by Th8;
hence  (L-1-Norm M) . x = (Lp-Norm (M,1)) . x by A3, A5, A7, POWER:25; ::_thesis: verum
end;
hence  L-1-Norm M = Lp-Norm (M,1) by A1, FUNCT_2:63; ::_thesis: verum
end;

theorem :: LPSPACE2:77
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds L-1-Space M = Lp-Space (M,1)
proof
let X be non empty set ; ::_thesis: for S being SigmaField of X
for M being sigma_Measure of S holds L-1-Space M = Lp-Space (M,1)
let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S holds L-1-Space M = Lp-Space (M,1)
let M be sigma_Measure of S; ::_thesis: L-1-Space M = Lp-Space (M,1)
 Pre-L-Space M = Pre-Lp-Space (M,1) by Th75;
hence  L-1-Space M = Lp-Space (M,1) by Th76; ::_thesis: verum
end;