 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 Th15:
  X\{0_No} c= Z & I1|Z = I2|Z implies
     divset(Y,o,X,I1) = divset(Y,o,X,I2)
proof
  assume
A1: X\{0_No} c= Z & I1|Z = I2|Z;
  thus divset(Y,o,X,I1) c= divset(Y,o,X,I2)
  proof
    let a;
    assume a in divset(Y,o,X,I1);
    then consider lamb be object such that
A2: lamb in Y & a in divs(lamb,o,X,I1) by Def3;
    divs(lamb,o,X,I1) = divs(lamb,o,X,I2) by A1,Th14;
    hence thesis by A2,Def3;
  end;
  let a;
  assume a in divset(Y,o,X,I2);
  then consider lamb be object such that
A3: lamb in Y & a in divs(lamb,o,X,I2) by Def3;
  divs(lamb,o,X,I1) =divs(lamb,o,X,I2) by A1,Th14;
  hence thesis by A3,Def3;
end;
