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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun3.html#D1" target="_self">PARTFUN3:def 1</a><br/>

<b>thus </b><a NAME="E2:13_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="partfun3.html#V4" target="_self">nonnegative-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun3.html#D4" target="_self">PARTFUN3:def 4</a><br/>

<b>thus </b><a NAME="E1:13_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> implies 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="partfun3.html#E1:13_1_1"><i><font color="Green">E5</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:13_1_1"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 ;<br/>

<a NAME="E4:13_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D9">RELAT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="partfun3.html#E1:13_1_1"><i><font color="Green">E5</font></i></a></i>;<br/>


</div>
