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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>



<b>consider </b><font color="Maroon" title="c5">S</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><br/><a NAME="E2:148"/><i><font color="Green" title="E83">A1</font></i>: 
 <a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <font color="Maroon" title="c2">b</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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">S</font>
 <b>and </b><br/><a NAME="E3:148"/><i><font color="Green" title="E84">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">S</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> )
 <b>by </b><i><a class="ref" href="polynom1.html#D18" target="_self" title="POLYNOM1:def.18">Def18</a></i>;<br/>
<a NAME="E4:148"/>
 <a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E84"><i><font color="Green" title="E70">Lm4</font></i></a></i>;<br/>
<a NAME="E5:148"/><b>then </b><i><font color="Green" title="E85">A3</font></i>: 
 <a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/>
<b>assume </b><a NAME="E6:148"/><i><font color="Green" title="E86">A4</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b2</font>
 ;<br/>

<a NAME="E7:148"/><b>then </b>
<font color="Maroon" title="c3">b1</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="ref" href="polynom1.html#T54" target="_self" title="POLYNOM1:th.54">Th54</a></i>;<br/>
<a NAME="E8:148"/><b>then </b>
<font color="Maroon" title="c3">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:148"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/>
<a NAME="E9:148"/><b>then </b>
<font color="Maroon" title="c3">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E2:148"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="polynom1.html#E5:148"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:148"/><i><font color="Green" title="E87">A5</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>and </b><br/><a NAME="E12:148"/><i><font color="Green" title="E88">A6</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">i</font>
;<br/>

<b>thus </b><a NAME="E13:148"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom1.html#E11:148"><i><font color="Green" title="E87">A5</font></i></a>, <a class="ref" href="polynom1.html#D19" target="_self" title="POLYNOM1:def.19">Def19</a></i>;<br/>

<a NAME="E14:148"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">b1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c3">b1</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E12:148"><i><font color="Green" title="E88">A6</font></i></a>, <a class="ref" href="polynom1.html#D19" target="_self" title="POLYNOM1:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E15:148"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom1.html#E6:148"><i><font color="Green" title="E86">A4</font></i></a>, <a class="ref" href="polynom1.html#T52" target="_self" title="POLYNOM1:th.52">Th52</a></i>;<br/>


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