 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 Th8:
  divset(Y,x,X,Inv) = divset(Y,x,X\{0_No},Inv)
proof
  thus divset(Y,x,X,Inv) c= divset(Y,x,X\{0_No},Inv)
  proof
    let o;
    assume o in divset(Y,x,X,Inv);
    then consider lamb be object such that
A1: lamb in Y & o in divs(lamb,x,X,Inv) by Def3;
    o in divs(lamb,x,X\{0_No},Inv) by A1,Th7;
    hence thesis by A1,Def3;
  end;
  let o;
  assume o in divset(Y,x,X\{0_No},Inv);
  then consider lamb be object such that
A2:lamb in Y & o in divs(lamb,x,X\{0_No},Inv) by Def3;
  o in divs(lamb,x,X,Inv) by A2,Th7;
  hence thesis by A2,Def3;
end;
