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<b>let </b><font color="Maroon">it1</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Maroon">it2</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>;<br/>

<b>given </b><font color="Maroon">S1</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span><b> such that </b><a NAME="E1:143_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S1</font>
 <b>and </b><br/><a NAME="E2:143_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font> iff <font color="Olive">p</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font> )
 ;<br/>

<b>given </b><font color="Maroon">S2</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span><b> such that </b><a NAME="E3:143_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">it2</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S2</font>
 <b>and </b><br/><a NAME="E4:143_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font> iff <font color="Olive">p</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:143_1_2_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> ;<br/><a NAME="E3:143_1_2_1_1"/>
<font color="Maroon">x'</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T12" target="_self">Th12</a>, </i>;<br/><b>hence </b><a NAME="E4:143_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T13" target="_self">Th13</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:143_1_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> ;<br/><a NAME="E4:143_1_2_1"/>
<font color="Maroon">x'</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T13" target="_self">Th13</a>, </i>;<br/><b>hence </b><a NAME="E5:143_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:143_1_2"/>
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T11" target="_self">Th11</a>, , <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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