reserve A,A1,A2,B,C,D for Ordinal,
  X,Y for set,
  x,y,a,b,c for object,
  L,L1,L2,L3 for Sequence,
  f for Function;
reserve fi,psi for Ordinal-Sequence;

theorem Th32:
  A in B implies C +^ A in C +^ B
proof
  defpred P[Ordinal] means A in $1 implies C +^ A in C +^ $1;
A1: for B st for D st D in B holds P[D] holds P[B]
  proof
    let B such that
A2: for D st D in B holds A in D implies C +^ A in C +^ D and
A3: A in B;
A4: now
      given D such that
A5:   B = succ D;
A6:   now
        assume A in D;
        then
A7:     C +^ A in C +^ D by A2,A5,ORDINAL1:6;
        succ(C +^ D) = C +^ succ D & C +^ D in succ(C +^ D) by Th28,ORDINAL1:6;
        hence thesis by A5,A7,ORDINAL1:10;
      end;
      now
        assume
A8:     not A in D;
        A c< D iff A c= D & A <> D;
        then C +^ A in succ(C +^ D) by A3,A5,A8,ORDINAL1:11,22;
        hence thesis by A5,Th28;
      end;
      hence thesis by A6;
    end;
    now
      deffunc F(Ordinal) = C+^$1;
      consider fi such that
A9:   dom fi = B & for D st D in B holds fi.D = F(D) from OSLambda;
      fi.A = C+^A by A3,A9;
      then
A10:  C+^A in rng fi by A3,A9,FUNCT_1:def 3;
      assume for D holds B <> succ D;
      then B is limit_ordinal by ORDINAL1:29;
      then C+^B = sup fi by A3,A9,Th29
        .= sup rng fi;
      hence thesis by A10,Th19;
    end;
    hence thesis by A4;
  end;
  for B holds P[B] from ORDINAL1:sch 2(A1);
  hence thesis;
end;
