reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;
reserve f1, f2 for non empty homogeneous to-naturals NAT*-defined Function,
  e1, e2 for homogeneous to-naturals NAT*-defined Function,
  p for Element of (arity f1+1)-tuples_on NAT;

theorem Th54:
  e1 is empty implies primrec(e1,e2,i) is empty
proof
  set f1 = e1, f2 = e2;
  assume
A1: f1 is empty;
  consider G being Function of (arity f1+1)-tuples_on NAT, HFuncs NAT such
  that
A2: primrec(f1,f2,i) = Union G and
A3: for p being Element of (arity f1+1)-tuples_on NAT holds G.p =
  primrec(f1,f2,i,p) by Def11;
A4: dom G = (arity f1+1)-tuples_on NAT by FUNCT_2:def 1;
  now
    set p = the Element of (arity f1+1)-tuples_on NAT;
    let y be object;
    hereby
      assume y in rng G;
      then consider x being object such that
A5:   x in dom G and
A6:   G.x = y by FUNCT_1:def 3;
      reconsider p = x as Element of (arity f1+1)-tuples_on NAT by A5;
      G.p = primrec(f1,f2,i,p) by A3
        .= {} by A1,Th52;
      hence y in {{}} by A6,TARSKI:def 1;
    end;
    assume y in {{}};
    then
A7: y = {} by TARSKI:def 1;
    G.p = primrec ( f1,f2,i,p) by A3
      .= {} by A1,Th52;
    hence y in rng G by A4,A7,FUNCT_1:3;
  end;
  then
A8: rng G = {{}} by TARSKI:2;
  Union G = {} by A8,ZFMISC_1:25;
  hence thesis by A2;
end;
