reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;
reserve n for Nat;

theorem  Th37:
  A c= B implies (unique_No_op B)| succ A = unique_No_op A
proof
  assume A c= B;
  then A1: A in succ B by ORDINAL1:22;
  then A2:succ A c= succ B by ORDINAL1:21;
  defpred P[Sequence,Ordinal,Surreal]  means
  $3 in union rng $1 or ($2 = born_eq $3 &
  ex Y be non empty surreal-membered set st
  Y=born_eq_set $3/\made_of union rng $1 & $3 = the Y -smallest Surreal);
  deffunc H(Sequence) = { e where e is Element of Day dom $1:
  for x st x=e holds P[$1,dom $1,x]};
  set S1=unique_No_op A,S = unique_No_op B,S2=S|succ A;
  A3:dom S1 = succ A by Def9;
  A4:dom S = succ B by Def9;
  then A5:dom S2 = succ A by A1,ORDINAL1:21,RELAT_1:62;
  A6: dom S1 = succ A & for B be Ordinal,L1 be Sequence st B in succ A &
  L1=S1|B holds S1.B = H(L1)
  proof
    thus dom S1 = succ A by Def9;
    let B be Ordinal,L1 be Sequence such that A7: B in succ A & L1=S1|B;
    A8:dom L1 = B by RELAT_1:62,A3,A7,ORDINAL1:def 2;
    A9: S1.B c= Day B by Def9,A7;
    thus S1.B c= H(L1)
    proof
      let o;
      assume A10: o in S1.B;
      then reconsider O=o as Surreal by A9;
      for x st x=o holds P[S1|B,dom (S1|B),x] by A10,Def9,A7,A8;
      hence thesis by A10,A8,A9,A7;
    end;
    let o;
    assume o in H(L1);
    then consider e be Element of Day dom L1 such that
    A11: o=e & for x st x=e holds P[L1,dom L1,x];
    P[L1,dom L1,e] by A11;
    hence thesis by A11,Def9,A7,A8;
  end;
  A12:dom S2 = succ A & for C be Ordinal,L2 be Sequence st C in succ A &
  L2=S2|C holds S2.C = H(L2)
  proof
    thus dom S2 = succ A by A4,A1,ORDINAL1:21,RELAT_1:62;
    let C be Ordinal,L1 be Sequence such that A13: C in succ A & L1=S2|C;
    A14:dom L1 = C by RELAT_1:62,A5,A13,ORDINAL1:def 2;
    A15: dom (S|C) = C by A13,A2,A4,RELAT_1:62,ORDINAL1:def 2;
    A16: S.C= S2.C by A13,FUNCT_1:49;
    A17: S.C c= Day C &
    for x holds x in S.C iff P[S|C,C,x] by A13,A2,Def9;
    A18: S2|C =(S|succ A)|C =S|C by A13,ORDINAL1:def 2,RELAT_1:74;
    thus S2.C c= H(L1)
    proof
      let o;
      assume A19:o in S2.C;
      then reconsider O=o as Surreal by A16,A17;
      for x st x=o holds P[S2|C,dom (S2|C),x] by A18,A16,A19,A15,A13,A2,Def9;
      hence thesis by A17,A19,A16,A13,A14;
    end;
    let o;
    assume o in H(L1);
    then consider e be Element of Day dom L1 such that
    A20: o=e & for x st x=e holds P[L1,dom L1,x];
    P[S|C,dom (S|C),e] by A20,A13,A18;
    hence thesis by A20,A16,A15,A13,A2,Def9;
  end;
  S1=S2 from ORDINAL1:sch 3(A6,A12);
  hence thesis;
end;
