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 Th14:
  for a, b be Complex holds
  (ComplexFuncAdd(A)). ((ComplexFuncExtMult(A)).[a,f],(
  ComplexFuncExtMult(A)).[b,f]) = (ComplexFuncExtMult(A)).[a+b,f]
proof
  let a, b be Complex;
  reconsider a, b as Element of COMPLEX by XCMPLX_0:def 2;
  reconsider ab = a+b as Element of COMPLEX by XCMPLX_0:def 2;
  now
    let x be Element of A;
    thus ((ComplexFuncAdd(A)). ((ComplexFuncExtMult(A)).[a,f],(
    ComplexFuncExtMult(A)).[b,f])).x = ((ComplexFuncExtMult(A)).[a,f]).x + ((
    ComplexFuncExtMult(A)).[b,f]).x by Th1
      .= a*(f.x) + ((ComplexFuncExtMult(A)).[b,f]).x by Th4
      .= a*(f.x) + b*(f.x) by Th4
      .= (a+b)*(f.x)
      .= ((ComplexFuncExtMult(A)).[ab,f]).x by Th4;
  end;
  hence thesis by FUNCT_2:63;
end;
