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 Th13:
  for a, b be Complex holds
  (ComplexFuncExtMult(A)).[a,(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 ((ComplexFuncExtMult(A)).[a,(ComplexFuncExtMult(A)).[b,f]]).x = a*(((
    ComplexFuncExtMult(A)).[b,f]).x) by Th4
      .= a*(b*(f.x)) by Th4
      .= (a*b)*(f.x)
      .= ((ComplexFuncExtMult(A)).[ab,f]).x by Th4;
  end;
  hence thesis by FUNCT_2:63;
end;
