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 v, u being VECTOR of (RLSp_L1Funct M) 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 v, u being VECTOR of (RLSp_L1Funct M) 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 v, u being VECTOR of (RLSp_L1Funct M) st f = v & g = u holds
f + g = v + u
let f, g be PartFunc of X,REAL ; :: thesis: for v, u being VECTOR of (RLSp_L1Funct M) st f = v & g = u holds
f + g = v + u
let v, u be VECTOR of (RLSp_L1Funct M); :: thesis: ( f = v & g = u implies f + g = v + u )
assume A1: ( f = v & g = u ) ; :: thesis: f + g = v + u
reconsider v2 = v as VECTOR of (RLSp_PFunct X) by TARSKI:def 3;
reconsider u2 = u as VECTOR of (RLSp_PFunct X) by TARSKI:def 3;
reconsider h = v2 + u2 as Element of PFuncs X,REAL ;
reconsider v3 = v2 as Element of PFuncs X,REAL ;
reconsider u3 = u2 as Element of PFuncs X,REAL ;
A3: ( dom h = (dom v3) /\ (dom u3) & ( for x being Element of X st x in dom h holds
h . x = (v3 . x) + (u3 . x) ) ) by Th10;
for x being set st x in dom h holds
h . x = (f . x) + (g . x) by A1, Th10;
then h = f + g by A1, A3, VALUED_1:def 1;
hence f + g = v + u by RLSUB121; :: thesis: verum