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<a NAME="E1:19_1_1"/><i><font color="Green">E11</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:19_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E2:19_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E3:19_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:19_1_1_1"/><i><font color="Green">E14</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="polynom3.html#E1:19_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="polynom3.html#E2:19_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E6:19_1_1_1"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E4:19_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="polynom3.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E7:19_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="polynom3.html#NR2" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="polynom3.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E8:19_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="polynom3.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E9:19_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="polynom3.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:19_1_1"/><b>then </b>
 <a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R6">is_connected_in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/>
<a NAME="E4:19_1_1"/><b>then </b>
 <a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V6">connected</a>
 
<b>by </b><i><a class="txt" href="polynom3.html#E1:19_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1"/>
 <a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">c<sub>1</sub></font> is <a href="orders_1.html#V3">being_linear-order</a>
 <b>by </b><i><a class="ref" href="orders_1.html#D5">ORDERS_1:def 5</a></i>;<br/>


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