reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;
reserve f1,f2 for Ordinal-Sequence;

theorem Th30:
  exp(C,A+^B) = exp(C,B)*^exp(C,A)
proof
  defpred P[Ordinal] means exp(C,A+^$1) = exp(C,$1)*^exp(C,A);
A1: 1 = exp(C,{}) by ORDINAL2:43;
A2: for B st P[B] holds P[succ B]
  proof
    let B such that
A3: exp(C,A+^B) = exp(C,B)*^exp(C,A);
    thus exp(C,A+^succ B) = exp(C,succ(A+^B)) by ORDINAL2:28
      .= C*^exp(C,A+^B) by ORDINAL2:44
      .= C*^exp(C,B)*^exp(C,A) by A3,ORDINAL3:50
      .= exp(C,succ B)*^exp(C,A) by ORDINAL2:44;
  end;
A4: for B st B <> 0 & B is limit_ordinal & for D st D in B holds P[D] holds
  P[B]
  proof
    deffunc F(Ordinal) = exp(C,$1);
    let B such that
A5: B <> 0 and
A6: B is limit_ordinal and
A7: for D st D in B holds exp(C,A+^D) = exp(C,D)*^exp(C,A);
    consider fi such that
A8: dom fi = B & for D st D in B holds fi.D = F(D) from ORDINAL2:sch
    3;
    consider psi such that
A9: dom psi = A+^B & for D st D in A+^B holds psi.D = F(D) from
    ORDINAL2:sch 3;
    deffunc F(Ordinal) = exp(C,$1);
    consider f1 such that
A10: dom f1 = A & for D st D in A holds f1.D = F(D) from ORDINAL2:sch
    3;
A11: now
      let D;
      assume D in dom(fi*^exp(C,A));
      then
A12:  D in dom fi by ORDINAL3:def 4;
      hence psi.((dom f1)+^D) = exp(C,A+^D) by A8,A9,A10,ORDINAL2:32
        .= (exp(C,D))*^exp(C,A) by A7,A8,A12
        .= (fi.D)*^exp(C,A) by A8,A12
        .= (fi*^exp(C,A)).D by A12,ORDINAL3:def 4;
    end;
A13: now
      let D such that
A14:  D in dom f1;
      A c= A+^B by ORDINAL3:24;
      hence psi.D = exp(C,D) by A9,A10,A14
        .= f1.D by A10,A14;
    end;
    dom psi = (dom f1)+^(dom(fi*^exp(C,A))) by A8,A9,A10,ORDINAL3:def 4;
    then
A15: psi = f1^(fi*^exp(C,A)) by A13,A11,Def1;
    exp(C,B)*^exp(C,A) is_limes_of fi*^exp(C,A) by A5,A6,A8,Th5,Th21;
    then
A16: exp(C,B)*^exp(C,A) is_limes_of psi by A15,Th3;
A17: A+^B <> {} by A5,ORDINAL3:26;
    A+^B is limit_ordinal by A5,A6,ORDINAL3:29;
    then lim psi = exp(C,A+^B) by A9,A17,ORDINAL2:45;
    hence thesis by A16,ORDINAL2:def 10;
  end;
  exp(C,A) = 1*^exp(C,A) by ORDINAL2:39;
  then
A18: P[0] by A1,ORDINAL2:27;
  for B holds P[B] from ORDINAL2:sch 1(A18,A2,A4);
  hence thesis;
end;
