reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem Th46:
  (A+^B)*^C = A*^C +^ B*^C
proof
  defpred S[Ordinal] means (A+^$1)*^C = A*^C +^ $1*^C;
A1: for B st S[B] holds S[succ B]
  proof
    let B such that
A2: (A+^B)*^C = A*^C +^ B*^C;
    thus (A+^succ B)*^C = (succ(A+^B))*^C by ORDINAL2:28
      .= A*^C +^ B*^C +^ C by A2,ORDINAL2:36
      .= A*^C +^ (B*^C +^ C) by Th30
      .= A*^C +^ (succ B)*^C by ORDINAL2:36;
  end;
A3: for B st B <> 0 & B is limit_ordinal & for D st D in B holds S[D] holds
  S[B]
  proof
    deffunc F(Ordinal) = A +^ $1;
    let B such that
A4: B <> 0 and
A5: B is limit_ordinal and
A6: for D st D in B holds S[D];
    consider fi such that
A7: dom fi = B & for D st D in B holds fi.D = F(D) from ORDINAL2:sch
    3;
    A+^B is limit_ordinal by A4,A5,Th29;
    then
A8: (A+^B)*^C is limit_ordinal by Th40;
A9: dom (fi*^C) = dom fi by Def4;
A10: now
      assume
A11:  C = {};
      then
A12:  A*^C = {} by ORDINAL2:38;
A13:  B*^C = {} by A11,ORDINAL2:38;
      (A+^B)*^C = {} by A11,ORDINAL2:38;
      hence thesis by A12,A13,ORDINAL2:27;
    end;
    deffunc F(Ordinal) = $1 *^ C;
    consider psi such that
A14: dom psi = B & for D st D in B holds psi.D = F(D) from ORDINAL2:
    sch 3;
A15: now
      let x be object;
      assume
A16:  x in B;
      then reconsider k = x as Ordinal;
      reconsider m = fi.k, n = psi.k as Ordinal;
      thus (fi*^C).x = m*^C by A7,A16,Def4
        .= (A+^k)*^C by A7,A16
        .= A*^C+^k*^C by A6,A16
        .= A*^C+^n by A14,A16
        .= (A*^C+^psi).x by A14,A16,Def1;
    end;
    reconsider k = psi.{} as Ordinal;
    {} in B by A4,Th8;
    then k in rng psi by A14,FUNCT_1:def 3;
    then
A17: k in sup rng psi by ORDINAL2:19;
    dom (A*^C+^psi) = dom psi by Def1;
    then
A18: fi*^C = A*^C+^psi by A7,A14,A9,A15,FUNCT_1:2;
A19: A+^B = sup fi by A4,A5,A7,ORDINAL2:29;
    now
      assume C <> {};
      then (A+^B)*^C = sup(fi*^C) by A4,A5,A7,A19,Th29,Th44
        .= A*^C+^sup psi by A4,A14,A18,Th43;
      hence (A+^B)*^C = union(A*^C+^sup psi) by A8
        .= A*^C+^union sup psi by A17,Th45
        .= A*^C+^B*^C by A4,A5,A14,ORDINAL2:37;
    end;
    hence thesis by A10;
  end;
  (A+^{})*^C = A*^C by ORDINAL2:27
    .= A*^C +^ {} by ORDINAL2:27
    .= A*^C +^ {}*^C by ORDINAL2:35;
  then
A20: S[0];
  for B holds S[B] from ORDINAL2:sch 1(A20,A1,A3);
  hence thesis;
end;
