reserve E, F, G,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 Th11:
  for X, Y be RealNormSpace
  for a be Real
  for v1, v2 be Lipschitzian LinearOperator of X,Y
  for w1, w2 be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
   st v1 = w1 & v2 = w2
  holds v1 + v2 = w1 + w2 & a(#)v1 = a * w1
proof
  let X, Y be RealNormSpace;
  let a be Real;
  let v1, v2 be Lipschitzian LinearOperator of X,Y;
  let zw1, zw2 be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
  assume A1: v1 = zw1 & v2 = zw2;

  reconsider zw12 = zw1 + zw2 as Point of
    R_NormSpace_of_BoundedLinearOperators(X,Y);
  zw12 is Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
  then A2: dom zw12 = the carrier of X by FUNCT_2:def 1;
  A3: dom(v1 + v2)
   = (dom v1) /\ (dom v2) by VFUNCT_1:def 1
  .= (the carrier of X) /\ (dom v2) by FUNCT_2:def 1
  .= (the carrier of X) /\ the carrier of X by FUNCT_2:def 1
  .= the carrier of X;

  for s be object st s in dom(v1 + v2)
  holds (v1 + v2).s = zw12.s
  proof
    let s be object;
    assume A4: s in dom (v1 + v2);
    then reconsider d = s as Point of X;
    thus (v1 + v2).s
     = (v1 + v2) /. d by A3,PARTFUN1:def 6
    .= (v1 /. d) + (v2 /. d) by A4,VFUNCT_1:def 1
    .= zw12.s by A1,LOPBAN_1:35;
  end;
  hence v1 + v2 = zw1 + zw2 by A2,A3,FUNCT_1:2;

  reconsider zw12 = a * zw1
    as Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
  zw12 is Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
  then  A5: dom zw12 = the carrier of X by FUNCT_2:def 1;
  A6: dom (a (#) v1)
   = dom v1 by VFUNCT_1:def 4
  .= the carrier of X by FUNCT_2:def 1;

  for s being object st s in dom(a (#) v1)
  holds (a (#) v1).s = zw12.s
  proof
    let s be object;
    assume A7: s in dom(a (#) v1);
    then reconsider d = s as Point of X;
    thus (a (#) v1).s
      = (a (#) v1) /. d by A6,PARTFUN1:def 6
    .= a * (v1 /. d) by A7,VFUNCT_1:def 4
    .= zw12.s by A1,LOPBAN_1:36;
  end;
  hence a (#) v1 = a * zw1 by A5,A6,FUNCT_1:2;
end;
