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;
