 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem Th5:
  Y is surreal-membered & X is surreal-membered &
     Inv is X -surreal-valued
implies
   divset(Y,x,X,Inv) is surreal-membered
proof
  assume
A1: Y is surreal-membered & X is surreal-membered &
  Inv is X -surreal-valued;
  let o;
  assume o in divset(Y,x,X,Inv);
  then consider l be object such that
A2:l in Y & o in divs(l,x,X,Inv) by Def3;
  reconsider l as Surreal by A1,A2;
  divs(l,x,X,Inv) is surreal-membered by Th4;
  hence thesis by A2;
end;
