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

theorem Th6:
  (FuncAdd(X,Y)).(f,(FuncExtMult(X,Y)).[-1,f]) = FuncZero(X,Y)
proof
  now
    let x be Element of X;
    set y=f.x;
    thus ((FuncAdd(X,Y)).(f,(FuncExtMult(X,Y)).[-jj,f])).x = f.x + ((
    FuncExtMult(X,Y)).[-jj,f]).x by Th1
      .= f.x + ((-1)*y) by Th2
      .= f.x + (-y) by RLVECT_1:16
      .= 0.Y by RLVECT_1:5
      .= (FuncZero(X,Y)).x by FUNCOP_1:7;
  end;
  hence thesis by FUNCT_2:63;
end;
