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_LpFunct (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_LpFunct (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_LpFunct (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_LpFunct (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_LpFunct (M,k)) st f = v & g = u holds

f + g = v + u

let v, u be VECTOR of (RLSp_LpFunct (M,k)); :: thesis: ( f = v & g = u implies f + g = v + u )

reconsider v2 = v, u2 = u as VECTOR of (RLSp_PFunct X) by TARSKI:def 3;

reconsider h = v2 + u2 as Element of PFuncs (X,REAL) ;

reconsider v2 = v2, u2 = u2 as Element of PFuncs (X,REAL) ;

assume A1: ( f = v & g = u ) ; :: thesis: f + g = v + u

A2: ( dom h = (dom v2) /\ (dom u2) & ( for x being Element of X st x in dom h holds

h . x = (v2 . x) + (u2 . x) ) ) by LPSPACE1:6;

for x being object st x in dom h holds

h . x = (f . x) + (g . x) by A1, LPSPACE1:6;

then h = f + g by A1, A2, VALUED_1:def 1;

hence f + g = v + u by LPSPACE1:4; :: thesis: verum

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_LpFunct (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_LpFunct (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_LpFunct (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_LpFunct (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_LpFunct (M,k)) st f = v & g = u holds

f + g = v + u

let v, u be VECTOR of (RLSp_LpFunct (M,k)); :: thesis: ( f = v & g = u implies f + g = v + u )

reconsider v2 = v, u2 = u as VECTOR of (RLSp_PFunct X) by TARSKI:def 3;

reconsider h = v2 + u2 as Element of PFuncs (X,REAL) ;

reconsider v2 = v2, u2 = u2 as Element of PFuncs (X,REAL) ;

assume A1: ( f = v & g = u ) ; :: thesis: f + g = v + u

A2: ( dom h = (dom v2) /\ (dom u2) & ( for x being Element of X st x in dom h holds

h . x = (v2 . x) + (u2 . x) ) ) by LPSPACE1:6;

for x being object st x in dom h holds

h . x = (f . x) + (g . x) by A1, LPSPACE1:6;

then h = f + g by A1, A2, VALUED_1:def 1;

hence f + g = v + u by LPSPACE1:4; :: thesis: verum