reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);
reserve v,u for VECTOR of RLSp_AlmostZeroLpFunct(M,k);

theorem Th48:
f in Lp_Functions(M,k) & g in Lp_Functions(M,k) & f a.e.= g,M implies
  Integral(M,(abs f) to_power k) = Integral(M,(abs g) to_power k)
proof
   set t = (abs f) to_power k;
   set s = (abs g) to_power k;
   assume
A1: f in Lp_Functions (M,k) & g in Lp_Functions (M,k) & f a.e.= g,M;
   then ex f1 be PartFunc of X,REAL st
    f=f1 & ex E be Element of S st M.E` =0 & dom f1 = E &
    f1 is E-measurable & (abs f1) to_power k is_integrable_on M; then
   consider Df be Element of S such that
A2: M.Df`=0 & dom f = Df & f is Df-measurable & t is_integrable_on M;
   ex g1 be PartFunc of X,REAL st
    g=g1 & ex E be Element of S st M.E` =0 & dom g1 = E &
    g1 is E-measurable & (abs g1) to_power k is_integrable_on M by A1; then
   consider Dg be Element of S such that
A3: M.Dg`=0 & dom g = Dg & g is Dg-measurable & s is_integrable_on M;
A4:dom(abs f) = dom f & dom(abs g) = dom g by VALUED_1:def 11;
   consider E1 being Element of S such that
A5: M.E1 = 0 & f|E1` = g|E1` by A1;
   reconsider NDf = Df`, NDg = Dg` as Element of S by MEASURE1:34;
   set Ef = Df \ (NDg \/ E1);
   set Eg = Dg \ (NDf \/ E1);
   set E2 = NDf \/ NDg \/ E1;
   NDf is measure_zero of M & NDg is measure_zero of M &
   E1 is measure_zero of M by A2,A3,A5,MEASURE1:def 7; then
   NDf \/ E1 is measure_zero of M & NDg \/ E1 is measure_zero of M
         by MEASURE1:37; then
A6:M.(NDf \/ E1) = 0 & M.(NDg \/ E1) = 0 by MEASURE1:def 7;
   X \ NDf = X /\ Df & X \ NDg = X /\ Dg by XBOOLE_1:48; then
A7:X \ NDf = Df & X \ NDg = Dg by XBOOLE_1:28;
   Ef = (Df \ NDg) \ E1 & Eg = (Dg \ NDf) \ E1 by XBOOLE_1:41; then
A8:Ef = (X \ (NDf \/ NDg)) \ E1 & Eg = (X \ (NDf \/ NDg)) \ E1
        by A7,XBOOLE_1:41; then
A9:Ef = X \ E2 & Eg = X \ E2 by XBOOLE_1:41;
   (abs f) is Df-measurable & (abs g) is Dg-measurable
     by A2,A3,MESFUNC6:48; then
A10:t is Df-measurable & s is Dg-measurable by A2,A3,A4,MESFUN6C:29;
A11:dom t = Df & dom s = Dg by A2,A3,A4,MESFUN6C:def 4; then
A12:Integral(M,t|Ef) = Integral(M,t) &
   Integral(M,s|Eg) = Integral(M,s) by A6,A10,MESFUNC6:89;
   dom(t|Ef) = dom t /\ Ef & dom(s|Ef) = dom s /\ Ef by RELAT_1:61; then
A13:dom(t|Ef) = (Df /\ Df) \ (NDg \/ E1) &
   dom(s|Ef) = (Dg /\ Dg) \ (NDf \/ E1) by A11,A8,XBOOLE_1:49;
   now let x be Element of X;
    assume A14: x in dom(t|Ef);
A15: dom(t|Ef) c= dom t & dom(s|Ef) c= dom s by RELAT_1:60;
    E2` c= E1` by XBOOLE_1:7,34; then
A16: f.x =(f|E1`).x & g.x = (g|E1`).x by A14,A13,A9,FUNCT_1:49;
    (t|Ef).x = t.x & (s|Ef).x = s.x by A14,A13,FUNCT_1:49; then
    (t|Ef).x = ((abs f).x) to_power k &
    (s|Ef).x = ((abs g).x) to_power k by A8,A13,A14,A15,MESFUN6C:def 4; then
    (t|Ef).x = (|.f.x qua Complex.|) to_power k &
    (s|Ef).x = (|.g.x qua Complex.|) to_power k
       by VALUED_1:18;
    hence (t|Ef).x = (s|Ef).x by A5,A16;
   end;
   hence thesis by A12,A13,A8,PARTFUN1:5;
end;
