 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 Th23:
  A c= B implies Positives A c= Positives B
proof
  assume
A1: A c= B;
  let x,y be object;
  assume
A2:[x,y] in Positives A;
  then reconsider xy=[x,y] as Surreal;
  xy in Day A c= Day B & 0_No < xy by A1,A2,Def10,SURREAL0:35;
  hence thesis by Def10;
end;
