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
  for V being add-associative right_zeroed right_complementable Abelian
    non empty addLoopStr
  for f1,f2,f3 being PartFunc of C,V holds
  f1 - (f2 + f3) = f1 - f2 - f3
proof
  let V be add-associative right_zeroed right_complementable Abelian
  non empty addLoopStr;
  let f1,f2,f3 be PartFunc of C,V;
A1: dom (f1 - (f2 + f3)) = dom f1 /\ dom (f2 + f3) by Def2
    .= dom f1 /\ (dom f2 /\ dom f3) by Def1
    .= dom f1 /\ dom f2 /\ dom f3 by XBOOLE_1:16
    .= dom (f1 - f2) /\ dom f3 by Def2
    .= dom (f1 - f2 - f3) by Def2;
  now
    let c;
    assume
A2: c in dom (f1 - (f2 + f3));
    then c in dom f1 /\ dom (f2 + f3) by Def2;
    then
A3: c in dom (f2 + f3) by XBOOLE_0:def 4;
    c in dom (f1 - f2) /\ dom f3 by A1,A2,Def2;
    then
A4: c in dom (f1 - f2) by XBOOLE_0:def 4;
    thus (f1 - (f2 + f3))/.c = (f1/.c) - ((f2 + f3)/.c) by A2,Def2
      .= (f1/.c) - ((f2/.c) + (f3/.c)) by A3,Def1
      .= (f1/.c) - (f2/.c) - (f3/.c) by RLVECT_1:27
      .= ((f1 - f2)/.c) - (f3/.c) by A4,Def2
      .= (f1 - f2 - f3)/.c by A1,A2,Def2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
