reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem OP2:
  for E be RealNormSpace,
    X,Y be Subset of E,
      v be Point of E
  st X is open & Y = {x-v where x is Point of E : x in X}
  holds Y is open
  proof
    let E be RealNormSpace,
        X,Y be Subset of E,
        v be Point of E;
    assume that
    A1: X is open and
    A2: Y = {x-v where x is Point of E : x in X};
    set w = -v;
    {x+w where x is Point of E : x in X} c= the carrier of E
    proof
      let s be object;
      assume s in {x+w where x is Point of E : x in X}; then
      ex x be Point of E st s=x+w & x in X;
      hence s in the carrier of E;
    end; then
    reconsider Z = {x+w where x is Point of E : x in X} as Subset of E;
    deffunc F(Point of E) = $1+w;
    deffunc G(Point of E) = $1-v;
    defpred P[Point of E] means $1 in X;
    A3: for v being Element of the carrier of E holds F(v) = G(v);
    {F(v1) where v1 is Element of the carrier of E : P[v1]}
      = {G(v2) where v2 is Element of the carrier of E : P[v2]}
        from FRAENKEL:sch 5 (A3);
    hence thesis by A1,A2,OP1;
  end;
