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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E4:9_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:9_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:9_1_1_1"/>
<font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:9_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E10:9_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E11:9_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E2:9_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:9_1_1"/>
<font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font> is <a href="seqm_3.html#V6">natural-yielding</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>


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