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 Th4:
  for V being Abelian non empty addLoopStr
  for f1,f2 being PartFunc of C,V holds
  f1 + f2 = f2 + f1
proof
  let V be Abelian non empty addLoopStr;
  let f1,f2 be PartFunc of C,V;
A1: dom (f1 + f2) = dom f2 /\ dom f1 by Def1
    .= dom (f2 + f1) by Def1;
  now
    let c;
    assume
A2: c in dom (f1+f2);
    hence (f1 + f2)/.c = (f2/.c) + (f1/.c) by Def1
      .= (f2 + f1)/.c by A1,A2,Def1;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
