 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 Th14:
  X\{0_No} c= Z & I1|Z = I2|Z implies
     divs(a,b,X,I1) = divs(a,b,X,I2)
proof
  assume
A1: X\{0_No} c= Z & I1|Z = I2|Z;
  thus divs(a,b,X,I1) c= divs(a,b,X,I2)
  proof
    let o;
    assume o in divs(a,b,X,I1);
    then consider xL be object such that
A2: xL in X & xL <> 0_No &
    o = (1_No +'(xL +' -' b) *' a) *' (I1.xL) by Def2;
A3: xL in X\{0_No} by A2,ZFMISC_1:56;
    then I1.xL  = (I2|Z).xL by A1,FUNCT_1:49
    .= I2.xL by A3,A1,FUNCT_1:49;
    hence thesis by A2,Def2;
  end;
  let o;
  assume o in divs(a,b,X,I2);
  then consider xL be object such that
A4:xL in X & xL <> 0_No &
  o = (1_No +'(xL +' -' b) *' a) *' (I2.xL) by Def2;
A5:xL in X\{0_No} by A4,ZFMISC_1:56;
  then I2.xL = (I1|Z).xL by A1,FUNCT_1:49
  .= I1.xL by A5,A1,FUNCT_1:49;
  hence thesis by A4,Def2;
end;
