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<b>let </b><font color="Maroon" title="c3">it1</font>, <font color="Maroon" title="c4">it2</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>;<br/>

<b>given </b><font color="Maroon" title="c5">S1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><a NAME="E1:143_1_2"/><i><font color="Green" title="E77">A1</font></i>: 
<font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">S1</font>
 <b>and </b><br/><a NAME="E2:143_1_2"/><i><font color="Green" title="E78">A2</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font> iff <font color="Olive" title="b1">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c2">b</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c6">S2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><a NAME="E3:143_1_2"/><i><font color="Green" title="E79">A3</font></i>: 
<font color="Maroon" title="c4">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c6">S2</font>
 <b>and </b><br/><a NAME="E4:143_1_2"/><i><font color="Green" title="E80">A4</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font> iff <font color="Olive" title="b1">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c2">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" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">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" title="E81">A5</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c7">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E3:143_1_2_1_1"/>
<font color="Maroon" title="c8">x'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:143_1_2"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="polynom1.html#E1:143_1_2_1_1"><i><font color="Green" title="E81">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E4:143_1_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E4:143_1_2"><i><font color="Green" title="E80">A4</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:143_1_2_1"/><i><font color="Green" title="E81">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c7">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E4:143_1_2_1"/>
<font color="Maroon" title="c8">x'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E4:143_1_2"><i><font color="Green" title="E80">A4</font></i></a>, <a class="txt" href="polynom1.html#E2:143_1_2_1"><i><font color="Green" title="E81">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E5:143_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:143_1_2"><i><font color="Green" title="E78">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:143_1_2"/>
<font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:143_1_2"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="polynom1.html#E3:143_1_2"><i><font color="Green" title="E79">A3</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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