reserve X, Y for RealNormSpace;

theorem
  for V,W be Subset of X, V1,W1 be Subset of LinearTopSpaceNorm X st V=
  V1 & W=W1 holds V+W=V1+W1
proof
  let V,W be Subset of X, V1,W1 be Subset of LinearTopSpaceNorm X such that
A1: V=V1 and
A2: W=W1;
  thus V+W c= V1+W1
  proof
    let x be object;
    assume x in V+W;
    then consider u,v be Point of X such that
A3: x=u+v and
A4: u in V and
A5: v in W;
    reconsider v1=v as Point of LinearTopSpaceNorm X by A2,A5;
    reconsider u1=u as Point of LinearTopSpaceNorm X by A1,A4;
    u+v=u1+v1 by NORMSP_2:def 4;
    hence thesis by A1,A2,A3,A4,A5;
  end;
  let x be object;
  assume x in V1+W1;
  then consider u,v be Point of LinearTopSpaceNorm X such that
A6: x=u+v and
A7: u in V1 and
A8: v in W1;
  reconsider v1=v as Point of X by A2,A8;
  reconsider u1=u as Point of X by A1,A7;
  u+v=u1+v1 by NORMSP_2:def 4;
  hence thesis by A1,A2,A6,A7,A8;
end;
