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
  A c= B & B in succ C implies (unique_No_op C).A c= (unique_No_op C).B
proof
  assume A1:A c= B & B in succ C;
  set MC=unique_No_op C;
  let o;
  assume A2: o in (unique_No_op C).A;
  then reconsider x=o as Surreal by SURREAL0:def 16;
  set b=born x;
  succ C =dom MC by Def9;
  then A3:dom (MC|B) = B by RELAT_1:62,A1,ORDINAL1:def 2;
  per cases;
  suppose A=B;
    hence thesis by A2;
  end;
  suppose A <>B;
    then A4:A in B by ORDINAL1:11,A1,XBOOLE_0:def 8;
    then MC.A = (MC|B).A by A3,FUNCT_1:47;
    then MC.A in rng (MC|B) by A3,A4,FUNCT_1:def 3;
    then o in union rng (MC|B) by A2,TARSKI:def 4;
    then x in MC.B by A1,Def9;
    hence thesis;
  end;
end;
