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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:43_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>2</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>and </b><br/><a NAME="E2:43_1_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>2</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1_2_1_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:43_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E4:43_1_2_1_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_2_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><a NAME="E5:43_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_2"><i><font color="Green">E28</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:43_1_2_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:43_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E5:43_1_2_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="bagorder.html#E2:43_1_2_1"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><a NAME="E6:43_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_2"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


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