
theorem
  for f be complex-valued Function holds abs f = delneg f + delpos f
  proof
    let f be complex-valued Function;
    reconsider fabs = abs f as (dom f)-defined complex-valued Function;
    reconsider g = (1/2)(#)f as (dom f)-defined complex-valued Function;
    reconsider h = g - g as total (dom f)-defined empty-yielding Function;
    delneg f + delpos f = (1/2)(#)fabs + (1/2)(#)f + (1/2)(#)(fabs - f)
      by RFUNCT_1:16
    .= (1/2)(#)fabs + (1/2)(#)f + ((1/2)(#)(fabs) - (1/2)(#)f) by RFUNCT_1:18
    .= (1/2)(#)fabs + (1/2)(#)f + (1/2)(#)fabs - (1/2)(#)f by RFUNCT_1:23
    .= (1/2)(#)f + ((1/2)(#)fabs + (1/2)(#)fabs) - (1/2)(#)f by RFUNCT_1:8
    .= (1/2)(#)f - (1/2)(#)f + ((1/2)(#)fabs + (1/2)(#)(fabs)) by RFUNCT_1:23
    .= ((1/2)(#)f - (1/2)(#)f) + ((1/2)+(1/2))(#)(fabs) by TOPREALC:2
    .= fabs;
    hence thesis;
  end;
