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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> 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="c4">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="E2:8"/><i><font color="Green" title="E9">A1</font></i>: 
(  <a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <font color="Maroon" title="c2">b</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="c4">S</font> &amp; ( 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="c4">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" title="POLYNOM1:def.18">POLYNOM1:def 18</a></i>;<br/>
<a NAME="E3:8"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/>
<a NAME="E4:8"/><b>then </b>
 <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="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a></i>;<br/>
<a NAME="E5:8"/><b>then </b><i><font color="Green" title="E10">A2</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="c4">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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:8_1"/>
<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>
 ;<br/><a NAME="E2:8_1"/><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="c4">S</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="hilbasis.html#E5:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:8_1"/>
<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="txt" href="hilbasis.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E7:8"/>
<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>
 ;<br/>

<a NAME="E8:8"/><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="c4">S</font>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:8"/>
<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="hilbasis.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="hilbasis.html#E5:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>


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