reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem
  for S being non empty addLoopStr for seq1,seq2 be sequence of S holds
  seq1 - seq2 = seq1 +- seq2
proof
  let S be non empty addLoopStr;
  let seq1, seq2 be sequence of S;
  for n being Element of NAT holds (seq1 - seq2).n = (seq1 +- seq2).n
  proof
    let n be Element of NAT;
    thus (seq1 - seq2).n = seq1.n - seq2.n by NORMSP_1:def 3
      .= seq1.n + (-seq2).n by BHSP_1:44
      .= (seq1+-seq2).n by NORMSP_1:def 2;
  end;
  hence thesis by FUNCT_2:63;
end;
