theorem Lem12:
  for F being Function-yielding Function, f being Function holds
  doms(F+*(a,f)) = (doms F)+*(a, dom f)
  proof
    let F be Function-yielding Function;
    let f be Function;
A1: dom doms(F+*(a,f)) = dom (F+*(a,f)) = dom F = dom doms F
    = dom((doms F)+*(a, dom f)) by FUNCT_6:def 2,FUNCT_7:30;
    hence dom doms(F+*(a,f)) = dom((doms F)+*(a, dom f));
    let b; assume
A2: b in dom doms(F+*(a,f));
    then
A3: doms(F+*(a,f)).b = dom ((F+*(a,f)).b) by A1,FUNCT_6:def 2;
    per cases;
    suppose
A4:   a = b;
      hence doms(F+*(a,f)).b = dom f by A1,A2,A3,FUNCT_7:31
      .= ((doms F)+*(a, dom f)).b by A1,A2,A4,FUNCT_7:31;
    end;
    suppose
A4:   a <> b;
      hence doms(F+*(a,f)).b = dom (F.b) by A3,FUNCT_7:32
      .= (doms F).b by A1,A2,FUNCT_6:def 2
      .= ((doms F)+*(a, dom f)).b by A4,FUNCT_7:32;
    end;
  end;
