reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem Th25:
  for V being non empty addLoopStr
  for f1,f2 being PartFunc of C,V holds
  f1 - f2 = f1 + -f2
proof
  let V be non empty addLoopStr;
  let f1,f2 be PartFunc of C,V;
A1: dom (f1 - f2) = dom f1 /\ dom f2 by Def2
    .= dom f1 /\ dom (-f2) by Def5
    .= dom (f1 + -f2) by Def1;
  now
    let c;
    assume
A2: c in dom (f1+-f2);
    then c in dom f1 /\ dom (-f2) by Def1;
    then
A3: c in dom (-f2) by XBOOLE_0:def 4;
    thus (f1 + -f2)/.c = (f1/.c) + ((-f2)/.c) by A2,Def1
      .= (f1/.c) - (f2/.c) by A3,Def5
      .= (f1-f2)/.c by A1,A2,Def2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
