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 ThB10, A1; :: thesis: verum