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 scalar-distributive scalar-unital vector-distributive
    non empty RLSStruct
  for f1,f2 being PartFunc of C,V holds
  f1-f2 = (-1)(#)(f2-f1)
proof
  let V be add-associative right_zeroed right_complementable
  Abelian scalar-distributive scalar-unital vector-distributive
  non empty RLSStruct;
  let f1,f2 be PartFunc of C,V;
A1: dom (f1 - f2) = dom f2 /\ dom f1 by Def2
    .= dom (f2 - f1) by Def2
    .= dom ((-1)(#)(f2 - f1)) by Def4;
  now
A2: dom (f1 - f2) = dom f2 /\ dom f1 by Def2
      .= dom (f2 - f1) by Def2;
    let c such that
A3: c in dom (f1-f2);
    thus (f1 - f2)/.c = (f1/.c) - (f2/.c) by A3,Def2
      .= -((f2/.c) - (f1/.c)) by RLVECT_1:33
      .= (-1)*((f2/.c) - (f1/.c)) by RLVECT_1:16
      .= (-1)*((f2 - f1)/.c) by A3,A2,Def2
      .= ((-1)(#)(f2 - f1))/.c by A1,A3,Def4;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
