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 Th14:
  for X, Y, Z be RealNormSpace,
      v1, v2 be Lipschitzian LinearOperator of X,Y,
      s1, s2 be Lipschitzian LinearOperator of Y,Z,
      a be Real
  holds s1*(v1 + v2) = s1*v1 + s1*v2
      & (s1 + s2)*v1 = s1*v1 + s2*v1
      & s1*(a(#)v1) = a(#)(s1*v1)
proof
  let X, Y, Z be RealNormSpace,
      v1, v2 be Lipschitzian LinearOperator of X,Y,
      s1, s2 be Lipschitzian LinearOperator of Y,Z,
      a be Real;
  reconsider pv1=v1, pv2=v2 as Point of
    R_NormSpace_of_BoundedLinearOperators(X,Y) by LOPBAN_1:def 9;

  reconsider ps1=s1, ps2=s2 as Point of
    R_NormSpace_of_BoundedLinearOperators (Y,Z) by LOPBAN_1:def 9;

  A1: v1 + v2 = pv1 + pv2 by Th11;
  A2: ps1*pv1 = s1*v1 by Th13;
  A3: ps1*pv2 = s1*v2 by Th13;
  A4: ps2*pv1 = s2*v1 by Th13;
  A5: s1 + s2 = ps1 + ps2 by Th11;
  A6: a(#)v1 = a*pv1 by LOPBAN13:30;
  A7: v1 + v2 is Lipschitzian LinearOperator of X,Y by A1,LOPBAN_1:def 9;
  A8: s1*v1 is Lipschitzian LinearOperator of X,Z by A2,LOPBAN_1:def 9;
  A9: s1*v2 is Lipschitzian LinearOperator of X,Z by A3,LOPBAN_1:def 9;
  A10: s1 + s2 is Lipschitzian LinearOperator of Y,Z by A5,LOPBAN_1:def 9;
  A11: s2*v1 is Lipschitzian LinearOperator of X,Z by A4,LOPBAN_1:def 9;
  A12: a(#)v1 is Lipschitzian LinearOperator of X,Y by A6,LOPBAN_1:def 9;

  thus s1*(v1 + v2)
   = ps1*(pv1 + pv2) by A1,A7,Th13
  .= ps1*pv1 + ps1*pv2 by LOPBAN13:19
  .= s1*v1 + s1*v2 by A2,A3,A8,A9,Th11;

  thus (s1 + s2)*v1
   = (ps1 + ps2)*pv1 by A5,Th13,A10
  .= ps1*pv1 + ps2*pv1 by LOPBAN13:20
  .= s1*v1 + s2*v1 by A2,A4,A8,A11,Th11;

  thus s1*(a(#)v1)
   = ps1*(a*pv1) by A6,A12,Th13
  .= (a*ps1)*pv1 by LOPBAN13:28
  .= a*(ps1*pv1) by LOPBAN13:28
  .= a(#)(s1*v1) by A8,Th13,LOPBAN13:30;
end;
