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 u being VECTOR of (RLSp_L1Funct M) st f = u holds
( u + ((- 1) * u) = (X --> 0 ) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in L1_Functions M & g in L1_Functions M & 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 u being VECTOR of (RLSp_L1Funct M) st f = u holds
( u + ((- 1) * u) = (X --> 0 ) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in L1_Functions M & g in L1_Functions M & 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 u being VECTOR of (RLSp_L1Funct M) st f = u holds
( u + ((- 1) * u) = (X --> 0 ) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in L1_Functions M & g in L1_Functions M & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
let f be PartFunc of X,REAL ; :: thesis: for u being VECTOR of (RLSp_L1Funct M) st f = u holds
( u + ((- 1) * u) = (X --> 0 ) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in L1_Functions M & g in L1_Functions M & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
let u be VECTOR of (RLSp_L1Funct M); :: thesis: ( f = u implies ( u + ((- 1) * u) = (X --> 0 ) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in L1_Functions M & g in L1_Functions M & 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;
reconsider h = u2 + ((- 1) * u2) as Element of PFuncs X,REAL ;
assume A1: f = u ; :: thesis: ( u + ((- 1) * u) = (X --> 0 ) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in L1_Functions M & g in L1_Functions M & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
then A2: ( h = (RealPFuncZero X) | (dom f) & (- 1) * u2 = (- 1) * u ) by Th22, RLSUB122;
hence u + ((- 1) * u) = (X --> 0 ) | (dom f) by RLSUB121; :: thesis: ex v, g being PartFunc of X,REAL st
( v in L1_Functions M & g in L1_Functions M & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M )
u + ((- 1) * u) in L1_Functions M ;
then consider v being PartFunc of X,REAL such that
A4: ( v = u + ((- 1) * u) & ex ND being Element of S st
( M . ND = 0 & dom v = ND ` & v is_integrable_on M ) ) ;
u in L1_Functions M ;
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_integrable_on M ) ) ;
then consider ND being Element of S such that
A6: ( M . ND = 0 & dom f = ND ` & f is_integrable_on M ) by A1;
set g = X --> 0 ;
A7: X --> 0 in L1_Functions M by LmDef1;
v | (ND ` ) = ((X --> 0 ) | (ND ` )) | (ND ` ) by RLSUB121, A2, A4, A6;
then v | (ND ` ) = (X --> 0 ) | (ND ` ) by FUNCT_1:82;
then v a.e.= X --> 0 ,M by Def2, A6;
hence ex v, g being PartFunc of X,REAL st
( v in L1_Functions M & g in L1_Functions M & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) by A4, A7; :: thesis: verum