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 Th16:
  (ComplexFuncMult(A)).((ComplexFuncExtMult(A)).[a,f],g) = (
  ComplexFuncExtMult(A)).[a,(ComplexFuncMult(A)).(f,g)]
proof
  reconsider a as Element of COMPLEX by XCMPLX_0:def 2;
  now
    let x be Element of A;
    thus ((ComplexFuncMult(A)).((ComplexFuncExtMult(A)).[a,f],g)).x
       = ((ComplexFuncExtMult(A)).[a,f]).x * g.x by Th2
      .= (a*(f.x)) * g.x by Th4
      .= a*(f.x * g.x)
      .= a*(((ComplexFuncMult(A)).(f,g)).x) by Th2
      .= ((ComplexFuncExtMult(A)).[a,(ComplexFuncMult(A)).(f,g)]).x
           by Th4;
  end;
  hence thesis by FUNCT_2:63;
end;
