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# For every <math>x, y \in X,</math> if <math>x\precsim y</math> then <math>x\precsim^* y</math> , and
# For every <math>x, y \in X,</math> if <math>x\prec y</math> then <math>x\prec^* y</math> . Here, <math>x\prec y</math> means "<math>x \precsim y</math> and not <math>y \precsim x</math>".
The difference between these definitions is only in condition 3. When the extension is a partial order, condition 3 is not required, since it follows from condition 2. ''Proof'': suppose that <math>x \precsim y</math> and not <math>y \precsim x</math>. By condition 2, <math>x\precsim^* y</math>. By reflexivity, "not <math>y \precsim x</math>" implies that <math>y\neq x</math>. Since '''<math>\precsim^*</math> '''is a partial order, <math>x\precsim^* y</math> and <math>y\neq x</math> imply "not <math>y\precsim^* x</math>". Therefore, <math>x\prec^* y</math>.
== Order-extension principle ==
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