reserve X for non empty set;
reserve Y for ComplexLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Complex;
reserve u,v,w for VECTOR of CLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th9:
  (FuncAdd(X,Y)).((FuncExtMult(X,Y)).[a,f],(FuncExtMult(X,Y)).[b,f
  ]) = (FuncExtMult(X,Y)).[a+b,f]
proof
  reconsider a1=a,b1=b, ab=a+b as Element of COMPLEX by XCMPLX_0:def 2;
  now
    let x be Element of X;
    thus ((FuncAdd(X,Y)).
    ((FuncExtMult(X,Y)).[a1,f],(FuncExtMult(X,Y)).[b1,f])).x =
    ((FuncExtMult(X,Y)).[a1,f]).x + ((FuncExtMult(X,Y)).[b1,f]).x by LOPBAN_1:1
      .= a1*(f.x) + ((FuncExtMult(X,Y)).[b1,f]).x by Th2
      .= a*(f.x) + b*(f.x) by Th2
      .= (a+b)*(f.x) by CLVECT_1:def 3
      .= ((FuncExtMult(X,Y)).[ab,f]).x by Th2;
  end;
  then
  (FuncAdd(X,Y)).((FuncExtMult(X,Y)).[a,f],(FuncExtMult(X,Y)).[b,f
  ]) = (FuncExtMult(X,Y)).[a+b,f] by FUNCT_2:63;
  hence thesis;
end;
