theorem Th1: ::: move eventually to VALUED_2
  for f1, f2 being PartFunc of REAL, REAL m holds
  f1-f2 = f1+-f2
  proof
    let f1, f2 be PartFunc of REAL, REAL m;
A1: dom(f1-f2) = dom f1 /\ dom f2 by VALUED_2:def 46;
A2: dom(f1+-f2) = dom f1 /\ dom -f2 by VALUED_2:def 45;
A3: dom -f2 = dom f2 by NFCONT_4:def 3;
    now
      let x be object;
      assume
A4:   x in dom(f1-f2); then
A5:   x in dom f2 by A1,XBOOLE_0:def 4; then
A6:   f2.x = f2/.x & (-f2).x = (-f2)/.x by A3,PARTFUN1:def 6;
      thus (f1-f2).x = f1.x - f2.x by A4,VALUED_2:def 46
      .= f1.x + (-f2).x by A3,A5,A6,NFCONT_4:def 3
      .= (f1+-f2).x by A1,A2,A3,A4,VALUED_2:def 45;
    end;
    hence thesis by A1,A2,A3,FUNCT_1:2;
  end;
