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
  for X, Y be RealNormSpace,
      v1, v2, v3 be Lipschitzian LinearOperator of X,Y,
      a,b be Real
  holds v1 + v2 = v2 + v1
      & (v1+v2)+v3 = v1+(v2+v3)
      & a(#)(v1 + v2) = a(#)v1 + a(#)v2
      & (a+b)(#)v1 = a(#)v1 + b(#)v1
      & (a*b)(#)v1 = a(#)(b(#)v1)
proof
  let X, Y be RealNormSpace,
      v1, v2, v3 be Lipschitzian LinearOperator of X,Y,
      a,b be Real;

  reconsider w1=v1, w2=v2, w3=v3 as Point of
    R_NormSpace_of_BoundedLinearOperators(X,Y)
    by LOPBAN_1:def 9;

  A1: v1 + v2 = w1 + w2 by Th11;
  A2: v2 + v3 = w2 + w3 by Th11;
  A3: a(#)v1 = a*w1 by LOPBAN13:30;
  A4: a(#)v2 = a*w2 by LOPBAN13:30;
  A5: b(#)v1 = b*w1 by LOPBAN13:30;

  A6: v1 + v2 is Lipschitzian LinearOperator of X,Y by A1,LOPBAN_1:def 9;
  A7: v2 + v3 is Lipschitzian LinearOperator of X,Y by A2,LOPBAN_1:def 9;
  A8: a(#)v1 is Lipschitzian LinearOperator of X,Y by A3,LOPBAN_1:def 9;
  A9: a(#)v2 is Lipschitzian LinearOperator of X,Y by A4,LOPBAN_1:def 9;
  A10: b(#)v1 is Lipschitzian LinearOperator of X,Y by A5,LOPBAN_1:def 9;

  thus v1 + v2 = v2 + v1;

  thus (v1+v2)+v3
     = (w1+w2)+w3 by A1,A6,Th11
    .= w1+(w2+w3) by RLVECT_1:def 3
    .= v1+(v2+v3) by A2,A7,Th11;

  thus a(#)(v1 + v2)
     = a*(w1 + w2) by A6,Th11,LOPBAN13:30
    .= a*w1 + a*w2 by RLVECT_1:def 5
    .= a(#)v1+ a(#)v2 by A3,A4,A8,A9,Th11;

  thus (a+b)(#)v1
     = (a+b)*w1 by LOPBAN13:30
    .= a*w1 + b*w1 by RLVECT_1:def 6
    .= a(#)v1 + b(#)v1 by A3,A5,A8,A10,Th11;

  thus (a*b)(#)v1
     = (a*b)*w1 by LOPBAN13:30
    .= a*(b*w1) by RLVECT_1:def 7
    .= a(#)(b(#)v1) by A5,A10,LOPBAN13:30;
end;
