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,COMPLEX
for v, u being VECTOR of (CLSp_AlmostZeroFunct 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,COMPLEX
for v, u being VECTOR of (CLSp_AlmostZeroFunct 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,COMPLEX
for v, u being VECTOR of (CLSp_AlmostZeroFunct M) st f = v & g = u holds
f + g = v + u
let f, g be PartFunc of X,COMPLEX; :: thesis: for v, u being VECTOR of (CLSp_AlmostZeroFunct M) st f = v & g = u holds
f + g = v + u
let v, u be VECTOR of (CLSp_AlmostZeroFunct 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, u2 = u as VECTOR of (CLSp_L1Funct M) by TARSKI:def 3;
thus v + u = v2 + u2 by ZFMISC_1:87, FUNCT_1:49
.= f + g by A1, Th19 ; :: thesis: verum