reserve x1,x2,z for set;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,COMPLEX);
reserve a,b for Complex;

theorem Th11:
  (ComplexFuncAdd(A)).(f,(ComplexFuncExtMult(A)).[-1r,f]) = ComplexFuncZero(A)
proof
  reconsider mj = -1r as Element of COMPLEX by XCMPLX_0:def 2;
  now
    let x be Element of A;
    set y=f.x;
    thus ((ComplexFuncAdd(A)).(f,(ComplexFuncExtMult(A)).[mj,f])).x
     = f.x + ((ComplexFuncExtMult(A)).[mj,f]).x by Th1
      .= f.x + (mj*y) by Th4
      .= (ComplexFuncZero(A)).x by COMPLEX1:def 4,FUNCOP_1:7;
  end;
  hence thesis by FUNCT_2:63;
end;
