reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;

theorem Th23:
  R_EAL(f+g)=R_EAL f + R_EAL g & R_EAL(f-g)=R_EAL f - R_EAL g &
  dom(R_EAL(f+g))= dom(R_EAL f) /\ dom(R_EAL g) & dom(R_EAL(f-g))= dom(R_EAL f)
  /\ dom(R_EAL g) & dom(R_EAL(f+g))= dom f /\ dom g & dom(R_EAL(f-g))= dom f /\
  dom g
proof
  dom(R_EAL f - R_EAL g) = dom(R_EAL f) /\ dom(R_EAL g) by MESFUNC2:2;
  then
A1: dom(R_EAL f - R_EAL g) = dom(f-g) by VALUED_1:12;
A2: now
    let x be object;
    assume
A3: x in dom(R_EAL f - R_EAL g);
    then (R_EAL f - R_EAL g).x =(R_EAL f).x - (R_EAL g).x by MESFUNC1:def 4
      .=f.x - g.x by SUPINF_2:3;
    hence (R_EAL f - R_EAL g).x =(f-g).x by A1,A3,VALUED_1:13;
  end;
  dom(R_EAL f + R_EAL g) = dom(R_EAL f) /\ dom(R_EAL g) by MESFUNC2:2;
  then
A4: dom(R_EAL f + R_EAL g) = dom(f+g) by VALUED_1:def 1;
  now
    let x be object;
    assume
A5: x in dom(R_EAL f + R_EAL g);
    then (R_EAL f + R_EAL g).x =(R_EAL f).x + (R_EAL g).x by MESFUNC1:def 3
      .=f.x + g.x by SUPINF_2:1;
    hence (R_EAL f + R_EAL g).x = (f+g).x by A4,A5,VALUED_1:def 1;
  end;
  hence thesis by A4,A1,A2,FUNCT_1:2,MESFUNC2:2;
end;
