reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;
reserve v,u for VECTOR of RLSp_L1Funct M;

theorem Th31:
  f a.e.= f1,M & g a.e.= g1,M implies (f+g) a.e.= (f1+g1),M
proof
  assume that
A1: f a.e.= f1,M and
A2: g a.e.= g1,M;
  consider EQ1 being Element of S such that
A3: M.(EQ1)=0 and
A4: f|EQ1` =f1|EQ1` by A1;
  consider EQ2 being Element of S such that
A5: M.(EQ2)=0 and
A6: g|EQ2` =g1|EQ2` by A2;
A7: (EQ1\/EQ2)` c= EQ1` by XBOOLE_1:7,34;
  then f|(EQ1\/EQ2)` =f1|EQ1`|(EQ1\/EQ2)` by A4,FUNCT_1:51;
  then
A8: f|(EQ1\/EQ2)` =f1|(EQ1\/EQ2)` by A7,FUNCT_1:51;
A9: (EQ1\/EQ2)` c= EQ2` by XBOOLE_1:7,34;
  then g|(EQ1\/EQ2)` =g1|EQ2`|(EQ1\/EQ2)` by A6,FUNCT_1:51
    .=g1|(EQ1\/EQ2)` by A9,FUNCT_1:51;
  then
A10: (f+g)|(EQ1\/EQ2)` =f1|(EQ1\/EQ2)` + g1|(EQ1\/EQ2)` by A8,RFUNCT_1:44
    .=(f1+g1)|(EQ1\/EQ2)` by RFUNCT_1:44;
  M.(EQ1 \/ EQ2) = 0 by A3,A5,Lm4;
  hence thesis by A10;
end;
