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) ;
set g = X --> 0;
u + ((- 1) * u) in L1_Functions M ;
then consider v being PartFunc of X,REAL such that
A1: v = u + ((- 1) * u) and
ex ND being Element of S st
( M . ND = 0 & dom v = ND ` & v is_integrable_on M ) ;
assume A2: 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 A3: h = (RealPFuncZero X) | (dom f) by Th16;
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
A4: M . ND = 0 and
A5: dom f = ND ` and
f is_integrable_on M by A2;
A6: (- 1) * u2 = (- 1) * u by Th5;
hence u + ((- 1) * u) = (X --> 0) | (dom f) by A3, Th4; :: 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 )
v | (ND `) = ((X --> 0) | (ND `)) | (ND `) by A3, A6, A1, A5, Th4;
then v | (ND `) = (X --> 0) | (ND `) by FUNCT_1:51;
then A7: v a.e.= X --> 0,M by A4;
X --> 0 in L1_Functions M by Lm3;
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 A1, A7; :: thesis: verum