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 st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds

f - g in Lp_Functions (M,k)

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 st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds

f - g in Lp_Functions (M,k)

let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL

for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds

f - g in Lp_Functions (M,k)

let f, g be PartFunc of X,REAL; :: thesis: for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds

f - g in Lp_Functions (M,k)

let k be positive Real; :: thesis: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) implies f - g in Lp_Functions (M,k) )

assume A1: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) ; :: thesis: f - g in Lp_Functions (M,k)

then (- 1) (#) g in Lp_Functions (M,k) by Th26;

hence f - g in Lp_Functions (M,k) by Th25, A1; :: thesis: verum

for M being sigma_Measure of S

for f, g being PartFunc of X,REAL

for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds

f - g in Lp_Functions (M,k)

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 st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds

f - g in Lp_Functions (M,k)

let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL

for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds

f - g in Lp_Functions (M,k)

let f, g be PartFunc of X,REAL; :: thesis: for k being positive Real st f in Lp_Functions (M,k) & g in Lp_Functions (M,k) holds

f - g in Lp_Functions (M,k)

let k be positive Real; :: thesis: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) implies f - g in Lp_Functions (M,k) )

assume A1: ( f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) ; :: thesis: f - g in Lp_Functions (M,k)

then (- 1) (#) g in Lp_Functions (M,k) by Th26;

hence f - g in Lp_Functions (M,k) by Th25, A1; :: thesis: verum