reserve Z for set;

theorem
  for n being Nat, Z,x being set, f,g being PartFunc of Z,REAL n
  st x in dom (f+g) holds (f+g)/.x = (f/.x) + (g/.x)
  proof
    let n be Nat;
    let Z,x be set;
    let f,g be PartFunc of Z,REAL n;
    assume
A1: x in dom (f+g);
    dom(f+g) = dom f /\ dom g by VALUED_2:def 45;
    then x in dom f & x in dom g by A1,XBOOLE_0:def 4;
    then
A2: f.x = f/.x & g.x = g/.x by PARTFUN1:def 6;
    thus (f+g)/.x = (f+g).x by A1,PARTFUN1:def 6
    .= (f/.x) + (g/.x) by A1,A2,VALUED_2:def 45;
  end;
