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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">b</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Olive" title="b2">b</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Olive" title="b2">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Maroon" title="c2">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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Olive" title="b1">b</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Olive" title="b1">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> implies <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c3">b</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:147"/>
<font color="Maroon" title="c2">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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E2:147"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A1</font></span>: 
( <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E3:147"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> is    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">pid</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<span class="kw">consider </span><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="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:147"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
 <a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K18" title="PRE_POLY:func.18">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">S</font>
 <span class="kw">and </span><br/><a NAME="E7:147"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">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="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D16" target="_self" title="PRE_POLY:def.16">Def18</a></span>;<br/>
<a NAME="E8:147"/>
 <a href="pre_poly.html#K18" title="PRE_POLY:func.18">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="pre_poly.html#E78"><span class="lab"><font color="Green" title="E59">Lm4</font></span></a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></span>;<br/>
<a NAME="E9:147"/><span class="kw">then </span>
<font color="Maroon" title="c4">pid</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E2:147"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="pre_poly.html#D2" target="_self" title="PRE_POLY:def.2">Def2T</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:147"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:147"><span class="lab"><font color="Green" title="E68">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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