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 Th6:
  (FuncAdd(X,Y)).(f,(FuncExtMult(X,Y)).[-1r,f]) = FuncZero(X,Y)
proof
     reconsider mj = -1r as Element of COMPLEX by XCMPLX_0:def 2;
  now
    let x be Element of X;
    set y=f.x;
    thus ((FuncAdd(X,Y)).(f,(FuncExtMult(X,Y)).[mj,f])).x
       = f.x + ((FuncExtMult(X,Y)).[mj,f]).x by LOPBAN_1:1
      .= f.x + ((-1r)*y) by Th2
      .= f.x + (-y) by CLVECT_1:3
      .= 0.Y by RLVECT_1:5
      .= (FuncZero(X,Y)).x by Th1;
  end;
   then (FuncAdd(X,Y)).(f,(FuncExtMult(X,Y)).[mj,f]) = FuncZero(X,Y)
           by FUNCT_2:63;
  hence thesis;
end;
