reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem Th17:
  1 |^|^ a = 1
  proof
    defpred P[Ordinal] means 1 |^|^ $1 = 1;
A1: P[0] by Th13;
A2: for a st P[a] holds P[succ a]
    proof
      let A be Ordinal;
      assume P[A];
      hence 1 |^|^ succ A = exp(1,1) by Th14
      .= 1 by ORDINAL2:46;
    end;
A3: for A being Ordinal st A <> 0 & A is limit_ordinal &
    for B being Ordinal st B in A holds P[B] holds P[A]
    proof
      let A be Ordinal such that
A4:   A <> 0 & A is limit_ordinal and
A5:   for B being Ordinal st B in A holds 1|^|^B = 1;
      deffunc F(Ordinal) = 1 |^|^ $1;
      consider fi being Ordinal-Sequence such that
A6:   dom fi = A & for B being Ordinal st B in A holds fi.B = F(B)
      from ORDINAL2:sch 3;
A7:   1 |^|^ A = lim fi by A4,A6,Th15;
A8:   rng fi c= { 1 }
      proof
        let x being object;
        assume x in rng fi; then
        consider y being object such that
A9:     y in dom fi & x = fi.y by FUNCT_1:def 3;
        reconsider y as Ordinal by A9;
        x = 1 |^|^ y by A6,A9
        .= 1 by A5,A6,A9;
        hence x in { 1 } by TARSKI:def 1;
      end;
      now
        thus {} <> 1;
        let b,c such that
A10:    b in 1 & 1 in c;
        set x = the Element of A;
        reconsider x as Ordinal;
        take D = x; thus D in dom fi by A4,A6;
        let E be Ordinal;
        assume D c= E & E in dom fi; then
        fi.E in rng fi by FUNCT_1:def 3;
        hence b in fi.E & fi.E in c by A8,A10,TARSKI:def 1;
      end; then
      1 is_limes_of fi by ORDINAL2:def 9;
      hence 1 |^|^ A = 1 by A7,ORDINAL2:def 10;
    end;
    for a holds P[a] from ORDINAL2:sch 1(A1,A2,A3);
    hence thesis;
  end;
